{"id":90153,"date":"2022-03-29T14:28:35","date_gmt":"2022-03-29T08:58:35","guid":{"rendered":"https:\/\/bigideasmathanswers.com\/?p=90153"},"modified":"2022-04-07T14:34:22","modified_gmt":"2022-04-07T09:04:22","slug":"envision-math-common-core-grade-6-answer-key-topic-3","status":"publish","type":"post","link":"https:\/\/bigideasmathanswers.com\/envision-math-common-core-grade-6-answer-key-topic-3\/","title":{"rendered":"Envision Math Common Core Grade 6 Answer Key Topic 3 Numeric And Algebraic Expressions"},"content":{"rendered":"

Practice with the help of enVision Math Common Core Grade 6 Answer Key<\/a> Topic 3 Numeric and Algebraic Expressions<\/strong> regularly and improve your accuracy in solving questions.<\/p>\n

Envision Math Common Core 6th Grade Answers Key Topic 3 Numeric And Algebraic Expressions<\/h2>\n

?Topic essential Question\u00a0<\/strong> What are expressions and how they can be written and evaluated?
\nAnswer:
\nExpressions\u00a0are\u00a0mathematical\u00a0statements that have a minimum of two terms containing numbers or variables, or both, connected by an operator in between.
\nTo evaluate an algebraic expression means\u00a0to find the value of the expression when the variable is replaced by\u00a0a given number. To evaluate an expression, we substitute the given number for the variable in the expression and then simplify the expression using the order of operations.<\/p>\n

TOPIC 3 enVision STEM Project<\/h3>\n

Did You Know?<\/strong>
\nThere are more than 600,000 bridges in the United States.<\/p>\n

\"Envision<\/p>\n

Your Task: Design a Bridge<\/strong>
\nSuppose the proposed ma\u00d7imum weight limit for a new bridge in your community is 100,000 pounds. How many and what types of vehicles can be allowed to cross the bridge? How can the weight on the bridge be controlled? You and your classmates will begin the engineering design process to understand the problem, do necessary research, and brainstorm solutions.<\/p>\n

Topic 3 GET READY!<\/h3>\n

Review What You Know!<\/strong><\/p>\n

Vocabulary<\/strong><\/p>\n

Choose the best term from the box to complete each definition.<\/strong><\/p>\n

composite number
\nformula
\nnumerical expression
\nprime number<\/p>\n

Question 1.
\nA ____ is a rule that uses symbols to relate two or more quantities.
\nAnswer:
\nWe know that,
\nA “Formula” is a rule that uses symbols to relate two or more quantities
\nHence, from the above,
\nWe can conclude that
\nThe best term that is suitable for the given definition is: Formula<\/p>\n

Question 2.
\nThe number 12 is a ____ because it has more than two factors.
\nAnswer:
\nWe know that,
\nThe number 12 is a “Composite number” because it has more than two factors.
\nHence, from the above,
\nWe can conclude that
\nThe best term that is suitable for the given definition is: Composite number<\/p>\n

Question 3.
\nA ___ is a mathematical phrase that includes numbers and at least one operation.
\nAnswer:
\nWe know that,
\nA “Numerical expression” is a mathematical phrase that includes numbers and at least one operation.
\nHence, from the above,
\nWe can conclude that
\nThe best term that is suitable for the given definition is: Numerical expression<\/p>\n

Perimeter and Area<\/strong>
\nUse the formulas P = 2l + 2w and A = l \u00d7 w, where l is the length and w is the width, to find the perimeter, P, and the area, A, of each figure.<\/p>\n

Question 4.
\n\"Envision
\nP = ___
\nA = ____
\nAnswer:
\nThe given figure is:
\n\"Envision
\nNow,
\nFrom the given figure,
\nWe can observe that
\nThe length of a rectangle (L) is: 13 cm
\nThe width of a rectangle (W) is: 13 cm
\nNow,
\nWe know that,
\nThe perimeter of a rectangle (P) is: 2 (L + W)
\nThe area of a rectangle (A) is: Length \u00d7 Width
\nSo,
\nP = 2 \u00d7 (13 + 13)
\n= 2 \u00d726
\n= 52 cm
\nA = 13 \u00d7 13
\n= 169 cm\u00b2
\nHence, from the above,
\nWe can conclude that
\nThe perimeter of the given rectangle is: 52 cm
\nThe area of the given rectangle is: 169 cm\u00b2<\/p>\n

Question 5.
\n\"Envision
\nP = ___
\nA = ____
\nAnswer:
\nThe given figure is:
\n\"Envision
\nNow,
\nFrom the given figure,
\nWe can observe that
\nThe length of a rectangle (L) is 5 in.
\nThe width of a rectangle (W) is 21 in.
\nNow,
\nWe know that,
\nThe perimeter of a rectangle (P) is: 2 (L + W)
\nThe area of a rectangle (A) is: Length \u00d7 Width
\nSo,
\nP = 2 \u00d7 (21 + 5)
\n= 2 \u00d726
\n= 52 in.
\nA = 21 \u00d7 5
\n= 105 in.\u00b2
\nHence, from the above,
\nWe can conclude that
\nThe perimeter of the given rectangle is 52 in.
\nThe area of the given rectangle is 105 in.\u00b2<\/p>\n

Question 6.
\n\"Envision
\nP = ___
\nA = ____
\nAnswer:
\nThe given figure is:
\n\"Envision
\nNow,
\nFrom the given figure,
\nWe can observe that
\nThe length of a rectangle (L) is 9 m
\nThe width of a rectangle (W) is 15 m
\nNow,
\nWe know that,
\nThe perimeter of a rectangle (P) is: 2 (L + W)
\nThe area of a rectangle (A) is: Length \u00d7 Width
\nSo,
\nP = 2 \u00d7 (9 + 15)
\n= 2 \u00d724
\n= 48 m
\nA = 9 \u00d7 15
\n= 135 m\u00b2
\nHence, from the above,
\nWe can conclude that
\nThe perimeter of the given rectangle is 48 m
\nThe area of the given rectangle is 135 m\u00b2<\/p>\n

Multiples<\/strong>
\nWrite the first five multiples of each number.<\/strong><\/p>\n

Question 7.
\n8
\nAnswer:
\nThe given number is: 8
\nNow,
\nWe know that,
\nA multiple is\u00a0a number that can be divided by another number a certain number of times without a remainder
\nHence, from the above,
\nWe can conclude that
\nThe first 5 multiples of 8 are: 8, 16, 24, 32, and 40<\/p>\n

Question 8.
\n9
\nAnswer:
\nThe given number is: 9
\nNow,
\nWe know that,
\nA multiple is\u00a0a number that can be divided by another number a certain number of times without a remainder
\nHence, from the above,
\nWe can conclude that
\nThe first 5 multiples of 9 are: 9, 18, 27, 36, and 45<\/p>\n

Question 9.
\n10
\nAnswer:
\nThe given number is: 10
\nNow,
\nWe know that,
\nA multiple is\u00a0a number that can be divided by another number a certain number of times without a remainder
\nHence, from the above,
\nWe can conclude that
\nThe first 5 multiples of 10 are: 10, 20, 30, 40, and 50<\/p>\n

Question 10.
\n6
\nAnswer:
\nThe given number is: 6
\nNow,
\nWe know that,
\nA multiple is\u00a0a number that can be divided by another number a certain number of times without a remainder
\nHence, from the above,
\nWe can conclude that
\nThe first 5 multiples of 6 are: 6, 12, 18, 24, and 30<\/p>\n

Question 11.
\n4
\nAnswer:
\nThe given number is: 4
\nNow,
\nWe know that,
\nA multiple is\u00a0a number that can be divided by another number a certain number of times without a remainder
\nHence, from the above,
\nWe can conclude that
\nThe first 5 multiples of 4 are: 4, 8, 12, 16, and 20<\/p>\n

Question 12.
\n3
\nAnswer:
\nThe given number is: 3
\nNow,
\nWe know that,
\nA multiple is\u00a0a number that can be divided by another number a certain number of times without a remainder
\nHence, from the above,
\nWe can conclude that
\nThe first 5 multiples of 3 are: 3, 6, 9, 12, and 15<\/p>\n

Factors<\/strong><\/p>\n

Question 13.
\nHow can you find the factors of 12 and 15? Explain.
\nAnswer:
\nThe given numbers are: 12 and 15
\nNow,
\nWe know that,
\nA factor is\u00a0a number that divides into another number exactly and without leaving a remainder
\nHence, from the above,
\nWe can conclude that
\nThe factors of 12 are: 1, 2, 3, 4, 6, and 12
\nThe factors of 15 are: 1, 3, 5, and 15<\/p>\n

Operations<\/strong><\/p>\n

Question 14.
\nHow are the terms difference, sum, quotient, and product alike?
\nAnswer:
\nSum:
\nThe sum is the result of adding two or more numbers.
\nProduct:
\nThe product of two or more numbers is the result of multiplying these numbers.
\nQuotient:
\nThe quotient of two numbers is the result of the division of these numbers
\nDifference:
\nThe word difference is the result of subtracting one number from another<\/p>\n

Language Development<\/strong>
\nWrite terms and phrases related to Numeric expressions and Algebraic expressions in the Venn diagram.
\n\"Envision<\/p>\n

In the box below, draw pictures to represent the terms and phrases in the overlap section of your diagram.<\/p>\n

Topic 3 PICK A PROJECT<\/strong><\/p>\n

PROJECT 3A<\/strong>
\nWhat is the most inspiring poem you have read?
\nPROJECT: WRITE AN ALGEBRAIC POEM
\n\"Envision<\/p>\n

PROJECT 3B<\/strong>
\nWhat equipment do you need to play your favorite sport?
\nPROJECT: PLAN A TEAM PURCHASE
\n\"Envision<\/p>\n

PROJECT 3C<\/strong>
\nWhere is the most interesting place you have gone swimming?
\nPROJECT: DESIGN A POOL PATIO
\n\"Envision<\/p>\n

PROJECT 3D<\/strong>
\nIn what ways have you seen exponents used in the real world?
\nPROJECT: CALCULATE WITH exPONENTS
\n\"Envision<\/p>\n

Lesson 3.1 Understand and Represent exponents<\/h3>\n

Solve & Discuss It!<\/strong><\/p>\n

ACTIVITY<\/strong><\/p>\n

Fold a sheet of paper in half. Record the number of sections you see when it is unfolded. Continue folding the paper in half 4 more times. Record the number of sections each time. Describe any patterns you see.
\n\"\"
\nAnswer:
\nIt is given that
\nFold a sheet of paper in half. Record the number of sections you see when it is unfolded. Continue folding the paper in half 4 more times.
\nNow,
\nThe given figure is:
\n\"\"
\nNow,
\nFrom the given figure,
\nWe can observe that,
\nWhen we fold a sheet of paper in half, there are 2 sections when we unfold the sheet of paper
\nSo,
\nThe number of sections when we fold the paper 4 more times in half = (The number of times a sheet of paper folded) \u00d7 (The number of sections present when a sheet of paper is unfolded)
\n= 4 \u00d7 2
\n= 8 sections
\nHence, from the above,
\nWe can conclude that
\nThe number of sections when we fold the paper 4 more times in half is: 8 sections<\/p>\n

Look for Relationships<\/strong> How are the number of sections related to the number of folds?
\nAnswer:
\nThe figure from the above problem is:
\n\"\"
\nNow,
\nWhen we observe the above figure,
\nWe can say that
\nThe number of sections present when we folded a paper in half is: 2 sections
\nSo,
\nThe number of sections = 2 \u00d7 (The number of folds)
\nHence, from the above,
\nWe can conclude that
\nThe relationship between the number of sections and the number of folds is:
\nThe number of sections = 2 \u00d7 (The number of folds)<\/p>\n

Focus on math practices<\/strong>
\nUse Structure<\/strong> How many sections will there be after 6 folds? 7 folds?
\nAnswer:
\nFrom the above problem,
\nWe know that,
\nThe number of sections = 2 \u00d7 (The number of folds)
\nSo,
\nThe number of sections when there are 6 folds = 2 \u00d7 6
\n= 12 sections
\nThe number of sections when there are 7 folds = 2 \u00d7 7
\n= 14 sections
\nHence, from the above,
\nWe can conclude that
\nThe number of sections when there are 6 folds is: 12 sections
\nThe number of sections when there are 7 folds is: 14 sections<\/p>\n

VISUAL LEARNING<\/strong><\/p>\n

? Essential Question<\/strong> How can you write and evaluate numbers with exponents?
\nAnswer:
\nYou can use the “Order of operations” to evaluate the expressions containing exponents.
\nFirst, evaluate anything in Parentheses or grouping symbols.
\nNext, look for Exponents, followed by Multiplication and Division (reading from left to right)
\nLastly, addition and subtraction (again, reading from left to right)<\/p>\n

Example 1
\n<\/strong>The expression 2 \u00d7 2 \u00d7 2 represents the number of cells after 1 hour if there is 1 cell at the start. How can you write this expression using exponents? How many cells will there be after 1 hour?<\/p>\n

Reasoning<\/strong> Repeated multiplication can be represented in more than one way.
\n\"Envision<\/p>\n

Some bacteria cells divide every 20 minutes to make 2 cells.<\/p>\n

You can use an exponent to write the repeated multiplication of a number.
\n\"Envision
\nA number that can be written using exponents is called a power.<\/p>\n

You can use repeated multiplication to evaluate or find the value of power.<\/p>\n

Multiply the first two factors, 2 \u00d7 2 = 4.
\n\"Envision
\nThere will be 8 cells after 1 hour.<\/p>\n

Try It!<\/strong>
\nThere are 2 \u00d7 2 \u00d7 2 \u00d7 2 \u00d7 2 \u00d7 2 \u00d7 2 \u00d7 2 \u00d7 2 bacteria cells after 3 hours. Write the repeated multiplication as power and then evaluate.
\nAnswer:
\nIt is given that
\nThere are 2 \u00d7 2 \u00d7 2 \u00d7 2 \u00d7 2 \u00d7 2 \u00d7 2 \u00d7 2 \u00d7 2 bacteria cells after 3 hours
\nNow,
\nWe know that,
\nThe number that is repeatedly multiplied is the “Base”
\nThe “Exponent” tells us how many times the base is used as a factor
\nA number that can be written using “Exponents” is called a “Power”
\nSo,
\nThe representation of the given repeated multiplication in the form of power is: 29<\/sup>
\nSo,
\n29<\/sup> = 25 <\/sup> \u00d7\u00a024<\/sup>
\n= 32 \u00d7 16
\n= 512 cells
\nHence, from the above,
\nWe can conclude that
\nThere are 512 bacteria cells present after 3 hours<\/p>\n

Convince Me!<\/strong> Why can you represent the number of cells after two hours as power 26<\/sup>?
\nAnswer:
\nFrom Example 1,
\nIt is given that
\nThe number of bacteria cells present in 1 hour is: 23<\/sup>
\nNow,
\nWe know that,
\nam<\/sup> \u00d7 an<\/sup> = am + n<\/sup>
\nSo,
\nThe number of bacteria cells present in 2 hours = 23<\/sup> \u00d7 23<\/sup>
\n= 26<\/sup> bacteria cells
\nHence, from the above,
\nWe can conclude that
\nWe can represent the number of cells after 2 hours as 2<\/p>\n

Try It!<\/strong>
\nEvaluate (\\(\\frac{1}{3}\\))\u00b3?
\nAnswer:
\nThe given expression is: (\\(\\frac{1}{3}\\))\u00b3
\nSo,
\n(\\(\\frac{1}{3}\\))\u00b3
\n= \\(\\frac{1}{3}\\) \u00d7 \\(\\frac{1}{3}\\) \u00d7 \\(\\frac{1}{3}\\)
\n= \\(\\frac{1 \u00d7 1}{3 \u00d7 3}\\) \u00d7 \\(\\frac{1}{3}\\)
\n= \\(\\frac{1}{9}\\) \u00d7 \\(\\frac{1}{3}\\)
\n= \\(\\frac{1 \u00d7 1}{9 \u00d7 3}\\)
\n= \\(\\frac{1}{27}\\)
\nHence, from the above,
\nWe can conclude that
\n(\\(\\frac{1}{3}\\))\u00b3 = \\(\\frac{1}{27}\\)<\/p>\n

Try It!<\/strong>
\nRafael calculated the foil as 1.8 \u00d7 104<\/sup> units thick. Evaluate Rafael’s expression.
\nAnswer:
\nIt is given that
\nRafael calculated the foil as 1.8 \u00d7 104<\/sup> units thick.
\nNow,
\nWe know that,
\n104<\/sup> = 102<\/sup> \u00d7 102<\/sup>
\n= 100 \u00d7 100
\n= 10,000
\nSo,
\n1.8 \u00d7 104<\/sup> = 18,000
\nHence, from the above,
\nWe can conclude that
\nRafael calculated the foil as 18,000 units thick<\/p>\n

KEY CONCEPT<\/strong>
\nYou can represent a repeated multiplication expression using an exponent.
\n\"Envision
\nYou can evaluate a power using repeated multiplication.
\n54<\/sup> = 5 \u00d7 5 \u00d7 5 \u00d7 5 = 625<\/p>\n

Do You Understand?<\/strong><\/p>\n

Question 1.
\n? Essential Question<\/strong> How can you write and evaluate numbers with exponents?
\nAnswer:
\nYou can use the “Order of operations” to evaluate the expressions containing exponents.
\nFirst, evaluate anything in Parentheses or grouping symbols.
\nNext, look for Exponents, followed by Multiplication and Division (reading from left to right)
\nLastly, addition and subtraction (again, reading from left to right)<\/p>\n

Question 2.
\nLook for Relationships<\/strong> How many times is 4 used as a factor in the expression 45<\/sup>? Write the numerical expression as repeated multiplication.
\nAnswer:
\nThe given expression is: 45<\/sup>
\nNow,
\nWe know that,
\nThe “Exponent” tells how many times the base is used as a factor
\nSo,
\nIn the given expression,
\n4 is used 5 times as a factor
\nSo,
\nThe representation of the given expression in the form of a repeated multiplication is:
\n4 \u00d7 4 \u00d7 4 \u00d7 4 \u00d7 4
\nHence, from the above,
\nWe can conclude that
\n4 is used 5 times as a factor
\nThe representation of the given expression in the form of a repeated multiplication is:
\n4 \u00d7 4 \u00d7 4 \u00d7 4 \u00d7 4<\/p>\n

Question 3.
\nBe Precise<\/strong> What is a power that has the same value as 18<\/sup>? Explain.
\nAnswer:
\nThe given expression is: 18<\/sup>
\nNow,
\nWe know that,
\nThe value of 1 raised to an exponent of any whole number is always equal to 1
\nSo,
\nThe power that has the same value as the given expression is: 1n<\/sup>
\nWhere,
\nn = 0, 1, 2,…..
\nHence, from the above,
\nWe can conclude that
\nThe power that has the same value as the given expression is: 1n<\/sup>
\nWhere,
\nn = 0, 1, 2,…..<\/p>\n

Question 4.
\nConstruct Arguments<\/strong> Does 2.5 \u00d7 100<\/sup> equal 0, 1, 2.5, or 25? Justify your answer.
\nAnswer:
\nThe given expression is: 2.5 \u00d7 100<\/sup>
\nNow,
\nWe know that,
\nAny nonzero number raised to an exponent of zero has a value of 1
\nSo,
\n2.5 \u00d7 100<\/sup> = 2.5 \u00d7 1
\n= 2.5
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: 2.5<\/p>\n

Question 5.
\nModel with Math<\/strong>
\nHow would you write (\\(\\frac{1}{2}\\))3<\/sup> as repeated multiplication?
\nAnswer:
\nThe given expression is: (\\(\\frac{1}{2}\\))3<\/sup>
\nNow,
\nWe know that,
\nThe number that is repeatedly multiplied is the “Base”
\nThe “Exponent” tells how many times the base is used as a factor
\nSo,
\nThe representation of the given expression as a repeated multiplication is:
\n\\(\\frac{1}{2}\\) \u00d7 \\(\\frac{1}{2}\\) \u00d7 \\(\\frac{1}{2}\\)
\nHence, from the above,
\nWe can conclude that
\nThe representation of the given expression as a repeated multiplication is:
\n\\(\\frac{1}{2}\\) \u00d7 \\(\\frac{1}{2}\\) \u00d7 \\(\\frac{1}{2}\\)<\/p>\n

Do You Know How?<\/strong><\/p>\n

Question 6.
\nWrite 81 as a repeated multiplication of 3s. Then write it as a power.
\nAnswer:
\nThe given number is: 81
\nNow,
\nWe know that,
\n81 = 9 \u00d7 9
\n= 3 \u00d7 3 \u00d7 3 \u00d7 3
\nNow,
\nWe know that,
\nThe number that is repeatedly multiplied is the “Base”
\nThe “Exponent” tells how many times the base is used as a factor
\nSo,
\nThe representation of 81 in the form of power is:
\n81 = 34<\/sup>
\nHence, from the above,
\nWe can conclude that
\nThe representation of 81 as a repeated multiplication is:
\n81 = 3 \u00d7 3 \u00d7 3 \u00d7 3
\nThe representation of 81 as power is:
\n81 = 34<\/sup><\/p>\n

Question 7.
\nWrite 125 as a repeated multiplication of 5s. Then write it as a power.
\nAnswer:
\nThe given number is: 125
\nNow,
\nWe know that,
\n125 = 25 \u00d7 5
\n= 5 \u00d7 5 \u00d7 5
\nNow,
\nWe know that,
\nThe number that is repeatedly multiplied is the “Base”
\nThe “Exponent” tells how many times the base is used as a factor
\nSo,
\nThe representation of 125 in the form of power is:
\n125 = 53<\/sup>
\nHence, from the above,
\nWe can conclude that
\nThe representation of 125 as a repeated multiplication is:
\n125 = 5 \u00d7 5 \u00d7 5
\nThe representation of 125 as power is:
\n125 = 53<\/sup><\/p>\n

Question 8.
\nWhat is 0.75 \u00d7 0.75 \u00d7 0.75 \u00d7 0.75 \u00d7 0.75 written as a power?
\nAnswer:
\nThe given expression is: 0.75 \u00d7 0.75 \u00d7 0.75 \u00d7 0.75 \u00d7 0.75
\nNow,
\nWe know that,
\nThe number that is repeatedly multiplied is the “Base”
\nThe “Exponent” tells how many times the base is used as a factor
\nSo,
\nThe representation of 0.75 \u00d7 0.75 \u00d7 0.75 \u00d7 0.75 \u00d7 0.75 in the form of power is:
\n0.75 \u00d7 0.75 \u00d7 0.75 \u00d7 0.75 \u00d7 0.75 = 0.755<\/sup>
\nHence, from the above,
\nWe can conclude that
\nThe representation of 0.75 \u00d7 0.75 \u00d7 0.75 \u00d7 0.75 \u00d7 0.75 as power is:
\n0.75 \u00d7 0.75 \u00d7 0.75 \u00d7 0.75 \u00d7 0.75 = 0.755<\/sup><\/p>\n

Question 9.
\nWhat is \\(\\frac{3}{8}\\) \u00d7 \\(\\frac{3}{8}\\) \u00d7 \\(\\frac{3}{8}\\) written as a power?
\nAnswer:
\nThe given expression is: \\(\\frac{3}{8}\\) \u00d7 \\(\\frac{3}{8}\\) \u00d7 \\(\\frac{3}{8}\\)
\nNow,
\nWe know that,
\nThe number that is repeatedly multiplied is the “Base”
\nThe “Exponent” tells how many times the base is used as a factor
\nSo,
\nThe representation of \\(\\frac{3}{8}\\) \u00d7 \\(\\frac{3}{8}\\) \u00d7 \\(\\frac{3}{8}\\) in the form of power is:
\n\\(\\frac{3}{8}\\) \u00d7 \\(\\frac{3}{8}\\) \u00d7 \\(\\frac{3}{8}\\) = (\\(\\frac{3}{8}\\))3<\/sup>
\nHence, from the above,
\nWe can conclude that
\nThe representation of \\(\\frac{3}{8}\\) \u00d7 \\(\\frac{3}{8}\\) \u00d7 \\(\\frac{3}{8}\\) as power is:
\n\\(\\frac{3}{8}\\) \u00d7 \\(\\frac{3}{8}\\) \u00d7 \\(\\frac{3}{8}\\) = (\\(\\frac{3}{8}\\))3<\/sup><\/p>\n

In 10-13, evaluate each power.<\/strong><\/p>\n

Question 10.
\n(\\(\\frac{1}{6}\\))2<\/sup>
\nAnswer:
\nThe given expression is: (\\(\\frac{1}{6}\\))2<\/sup>
\nSo,
\n(\\(\\frac{1}{6}\\))2<\/sup>
\n= \\(\\frac{1}{6}\\) \u00d7 \\(\\frac{1}{6}\\)
\n= \\(\\frac{1 \u00d7 1}{6 \u00d7 6}\\)
\n= \\(\\frac{1}{36}\\)
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: \\(\\frac{1}{36}\\)<\/p>\n

Question 11.
\n450<\/sup>
\nAnswer:
\nThe given expression is: 450<\/sup>
\nNow,
\nWe know that,
\nAny nonzero number raised to an exponent of zero has a value of 1
\nSo,
\n450<\/sup> = 1
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: 1<\/p>\n

Question 12.
\n0.15<\/sup>
\nAnswer:
\nThe given given expression is: 0.15<\/sup>
\nSo,
\n0.15<\/sup>
\n= 0.1 \u00d7 0.1 \u00d70.1 \u00d7 0.1 \u00d7 0.1
\n= 0.00001
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: 0.00001<\/p>\n

Question 13.
\n73<\/sup>
\nAnswer:
\nThe given expression is: 73<\/sup>
\nSo,
\n73<\/sup>
\n= 7 \u00d7 7 \u00d7 7
\n= 49 \u00d7 7
\n= 343
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: 343<\/p>\n

In 14-16, evaluate each expression.<\/strong><\/p>\n

Question 14.
\n4.5 \u00d7 104<\/sup>
\nAnswer:
\nThe given expression is: 4.5 \u00d7 104<\/sup>
\nNow,
\nWe know that,
\n104<\/sup> = 10,000
\nSo,
\n4.5 \u00d7 104<\/sup>
\n= 45 \u00d7 103<\/sup>
\n= 45,000
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: 45,000<\/p>\n

Question 15.
\n0.6 \u00d7 106<\/sup>
\nAnswer:
\nThe given expression is: 0.6 \u00d7 106<\/sup>
\nNow,
\nWe know that,
\n106<\/sup> = 10,00,000
\nSo,
\n0.6 \u00d7 106<\/sup>
\n= 6 \u00d7 105<\/sup>
\n= 6,00,000
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: 6,00,000<\/p>\n

Question 16.
\n3.4 \u00d7 100<\/sup>
\nAnswer:
\nThe given expression is: 3.4 \u00d7 100<\/sup>
\nNow,
\nWe know that,
\nAny nonzero value raised to an exponent of zero has a value of 1
\nSo,
\n100<\/sup> = 1
\nSo,
\n3.4 \u00d7 100<\/sup>
\n= 3.4 \u00d7 1
\n= 3.4
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: 3.4<\/p>\n

Practice & Problem Solving<\/strong><\/p>\n

Scan for Multimedia<\/strong><\/p>\n

In 17-20, write the exponent for each expression.<\/strong><\/p>\n

Question 17.
\n9 \u00d7 9 \u00d7 9 \u00d7 9
\nAnswer:
\nThe given expression is: 9 \u00d7 9 \u00d7 9 \u00d7 9
\nNow,
\nWe know that,
\nThe “Exponent” tells how many times the base is used as a factor
\nSo,
\nFrom the given expression,
\nWe can observe that
\n9 is repeated 4 times
\nHence, from the above,
\nWe can conclude that
\nThe exponent for the given expression is: 4<\/p>\n

Question 18.
\n1.29<\/sup>
\nAnswer:
\nThe given expression is: 1.29<\/sup>
\nNow,
\nWe know that,
\nThe “Exponent” tells how many times the base is used as a factor
\nSo,
\nFrom the given expression,
\nWe can observe that
\n1.2 is repeated 9 times
\nHence, from the above,
\nWe can conclude that
\nThe exponent for the given expression is: 9<\/p>\n

Question 19.
\n\\(\\frac{1}{6}\\) \u00d7 \\(\\frac{1}{6}\\) \u00d7 \\(\\frac{1}{6}\\)
\nAnswer:
\nThe given expression is: \\(\\frac{1}{6}\\) \u00d7 \\(\\frac{1}{6}\\) \u00d7 \\(\\frac{1}{6}\\)
\nNow,
\nWe know that,
\nThe “Exponent” tells how many times the base is used as a factor
\nSo,
\nFrom the given expression,
\nWe can observe that
\n\\(\\frac{1}{6}\\) is repeated 3 times
\nHence, from the above,
\nWe can conclude that
\nThe exponent for the given expression is: 3<\/p>\n

Question 20.
\n7
\nAnswer:
\nThe given expression is: 7
\nNow,
\nWe know that,
\nThe “Exponent” tells how many times the base is used as a factor
\nSo,
\nFrom the given expression,
\nWe can observe that
\n7 is repeated 1 time
\nHence, from the above,
\nWe can conclude that
\nThe exponent for the given expression is: 1<\/p>\n

Leveled Practice<\/strong> In 21-26, evaluate each power or expression.<\/strong><\/p>\n

Question 21.
\n\"Envision
\nAnswer:
\nThe given expression is: 83<\/sup>
\nSo,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: 512<\/p>\n

Question 22.
\n\"Envision
\nAnswer:
\nThe given expression is: (\\(\\frac{1}{5}\\))4<\/sup>
\nSo,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: \\(\\frac{1}{625}\\)<\/p>\n

Question 23.
\n0.62<\/sup>
\nAnswer:
\nThe given expression is 0.6\u00b2
\nSo,
\n0.6\u00b2 = 0.6 \u00d7 0.6
\n= 0.36
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: 0.36<\/p>\n

Question 24.
\n(\\(\\frac{1}{4}\\))2<\/sup>
\nAnswer:
\nThe given expression is: (\\(\\frac{1}{4}\\))\u00b2
\nSo,
\n(\\(\\frac{1}{4}\\))\u00b2 = \\(\\frac{1}{4}\\) \u00d7 \\(\\frac{1}{4}\\)
\n= \\(\\frac{1 \u00d7 1}{4 \u00d7 4}\\)
\n= \\(\\frac{1}{16}\\)
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: \\(\\frac{1}{16}\\)<\/p>\n

Question 25.
\n580<\/sup>
\nAnswer:
\nThe given expression is: 580<\/sup>
\nNow,
\nWe know that,
\nAny nonzero value raised to an exponent of zero has a value of 1
\nSo,
\n580<\/sup> = 1
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: 1<\/p>\n

Question 26.
\n6.2 \u00d7 103<\/sup>
\nAnswer:
\nThe given expression is: 6.2 \u00d7 103<\/sup>
\nSo,
\n6.2 \u00d7 103<\/sup> = 62 \u00d7102<\/sup>
\n= 62 \u00d7 100
\n= 6,200
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: 6,200<\/p>\n

Question 27.
\nA company rents two storage units. Both units are cube-shaped. What is the difference in the volume between the two storage units? Note that the volume of a cube is s3<\/sup>, where s is the side length. Explain.
\n\"Envision
\nAnswer:
\nIt is given that
\nA company rents two storage units. Both units are cube-shaped
\nNow,
\nThe given figure is:
\n\"Envision
\nNow,
\nFrom the given figure,
\nWe can observe that
\nThe side length of the first storage unit is: 6.5 ft
\nThe side length of the second storage unit is: 6 ft
\nNow,
\nWe know that,
\nThe volume of a cube = (Side)\u00b3
\nSo,
\nThe difference in volume between the two storage units = (The volume of the first storage unit) – (The volume of the second storage unit)
\n= (6.5)\u00b3 – 6\u00b3
\n= 274.625 – 216
\n= 58.63 ft\u00b3
\nHence, from the above,
\nWe can conclude that
\nThe difference in volume between the two storage units is: 58.63 ft\u00b3<\/p>\n

Question 28.
\nJia is tiling a floor. The floor is a square with a side length of 12 feet. Jia wants the tiles to be squares with a side length of 2 feet. How many tiles does Jia need to cover the entire floor? Note that the area of a square is s2<\/sup>, where s is the side length. Explain.
\nAnswer:
\nIt is given that
\nJia is tiling a floor. The floor is a square with a side length of 12 feet. Jia wants the tiles to be squares with a side length of 2 feet.
\nNow,
\nWe know that,
\nThe area of a square = (Side)\u00b2
\nSo,
\nThe area of the square floor = (The sidelength of the square floor)\u00b2
\n= 12\u00b2
\n= 144 feet\u00b2
\nSo,
\nThe area of the square tiles = (The sidelength of the square tiles)\u00b2
\n= 2\u00b2
\n= 4 feet\u00b2
\nNow,
\nThe number of tiles Jia needed to cover the entire floor = (The area of the square floor) \u00f7 (The area of the square tiles)
\n= \\(\\frac{144}{4}\\)
\n= 36 tiles
\nHence, from the above,
\nWe can conclude that
\nThe number of tiles Jia needed to cover the entire floor is: 36 tiles<\/p>\n

Question 29.
\nA marine biologist studies the population of seals in a research area. How many seals are in the research area?
\n\"Envision
\nAnswer:
\nIt is given that
\nA marine biologist studies the population of seals in a research area
\nNow,
\nThe given figure is:
\n\"Envision
\nNow,
\nFrom the given figure,
\nWe can observe that
\nThe population of Seals is: 3.27 \u00d7 10\u00b2
\nNow,
\nWe know that,
\n10\u00b2 = 10 \u00d7 10
\n= 100
\nSo,
\n3.27 \u00d7 10\u00b2 = 32.7 \u00d7 10
\n= 327 Seals
\nHence, from the above,
\nWe can conclude that
\nThe number of Seals that are present in the research area is: 327 Seals<\/p>\n

Question 30.
\nHigher-Order Thinking<\/strong> Zach invested $50 and tripled his money in two years. Kayla also invested $50, and after two years the amount was equal to 50 to the third power. Who had more money after two years? Explain.
\nAnswer:
\nIt is given that
\nZach invested $50 and tripled his money in two years. Kayla also invested $50, and after two years the amount was equal to 50 to the third power
\nSo,
\nAccording to the given information,
\nThe amount of money invested by Zach in two years = $50 \u00d7 3
\n= $150
\nThe amount of money invested by Kayla in two years = ($50)\u00b3
\n= $1,25,000
\nSo,
\nFrom the above values,
\nWe can say that
\n$1,25,000 > $150
\nHence, from the above,
\nWe can conclude that
\nKayla had more money after two years<\/p>\n

Question 31.
\nMalik read that the land area of Alaska is about 5.7 \u00d7 105<\/sup> square miles. About how many square miles is the land area of Alaska?
\n\"Envision
\nAnswer:
\nIt is given that
\nMalik read that the land area of Alaska is about 5.7 \u00d7 105<\/sup> square miles.
\nNow,
\nWe know that,
\n105<\/sup> = 1,00,000
\n104<\/sup> = 10,000
\nSo,
\n5.7 \u00d7 105<\/sup> = 57 \u00d7 104<\/sup>
\n= 57 \u00d7 10,000
\n= 57,000 square miles
\nHence, from the above,
\nWe can conclude that
\nThe land area of Alaska is about 57,000 square miles<\/p>\n

Question 32.
\nExplain why the expressions 100<\/sup>, 14<\/sup>, and 1 \u00d7 1.00<\/sup> have the same value.
\nAnswer:
\nThe given expressions are: 100<\/sup>, 14<\/sup>, 1 \u00d7 1.04<\/sup>
\nNow,
\nWe know that,
\nAny nonzero value that has raised to an exponent of zero has a value of 1
\nAny value of 1 that is raised to an exponent that contained a whole number has always a value of 1
\nHence, from the above,
\nWe can conclude that
\nThe given expressions have the same value since some expressions have raised to an exponent of zero and some expressions have the value of 1 that is to an exponent of any whole number<\/p>\n

Question 33.
\nSolve the equation 0.33<\/sup> = n.
\nAnswer:
\nThe given equation is:
\n0.3\u00b3 = n
\nSo,
\nBy using the repeated multiplication,
\nn = 0.3 \u00d7 0.3 \u00d7 0.3
\nn = 0.09 \u00d7 0.3
\nn = 0.027
\nHence, from the above,
\nWe can conclude that
\nThe value of “n” for the given equation is: 0.027<\/p>\n

Question 34.
\nConstruct Arguments<\/strong> The same digits are used for the expressions 25<\/sup> and 52<\/sup>. Explain how to compare the values of the expressions.
\nAnswer:
\nIt is given that
\nThe same digits are used for the expressions 25<\/sup> and 52<\/sup>
\nNow,
\nWe know that,
\nThe number that is repeatedly multiplied is the “Base”
\nThe “Exponent” tells how many times the base is used as a factor
\nSo,
\n25<\/sup> = 2 \u00d7 2 \u00d7 2 \u00d7 2 \u00d7 2
\n= 4 \u00d7 4 \u00d7 2
\n= 16 \u00d7 2
\n= 32
\nSo,
\n52<\/sup> = 5 \u00d75
\n= 25
\nSo,
\nFrom the obtained values,
\n32 > 25
\nHence, from the above,
\nWe can conclude that 25<\/sup> is greater than 52<\/sup><\/p>\n

Question 35.
\nCritique Reasoning<\/strong> Kristen was asked to write each of the numbers in the expression 80,000 \u00d7 25 using exponents. Her response was (8 \u00d7 103<\/sup>) \u00d7 52<\/sup>. Was Kristen’s response correct? Explain.
\nAnswer:
\nIt is given that
\nKristen was asked to write each of the numbers in the expression 80,000 \u00d7 25 using exponents. Her response was (8 \u00d7 103<\/sup>) \u00d7 52<\/sup>
\nNow,
\nThe given expression is: 80,000 \u00d7 25
\nNow,
\n80,000 \u00d7 25 = 8 \u00d7 10,000 \u00d7 25
\n= 2\u00b3 \u00d7 10,000 \u00d7 5\u00b2
\nNow,
\nWe know that,
\n10,000 = 104<\/sup>
\nSo,
\n80,000 \u00d7 25 = 23<\/sup> \u00d7 4<\/sup> \u00d7 52<\/sup>
\n= (8 \u00d7 104<\/sup>) \u00d7 52<\/sup>
\nHence, from the above,
\nWe can conclude that
\nKristen’s response was not correct<\/p>\n

Question 36.
\nConsider the equation 1,000,000 = 106<\/sup>. Why is 10 used as the base to write 106<\/sup>?
\nAnswer:
\nThe given equation is: 1,000,000 = 106<\/sup>
\nNow,
\nWe know that,
\nIn base-10, each digit of a number can have an integer value ranging from 0 to 9 (10 possibilities) depending on its position. The places or positions of the numbers are based on powers of 10. Each number position is 10 times the value to the right of it, hence the term base-10
\nHence, from the above,
\nWe can conclude that
\nWe used 10 as the base to write 106<\/sup> because of the position and the possibilities<\/p>\n

Question 37.
\nIsabella saved 2 nickels today. If she doubles the number of nickels she saves each day, how many days, including today, will it take her to save more than 500 nickels?
\nAnswer:
\nIt is given that
\nIsabella saved 2 nickels today and she doubles the number of nickels she saves each day,
\nSo,
\nThe number of nickels she saved today is: 2
\nThe number of nickels she saved the second day is 4 (or) 2\u00b2
\nThe number of nickels she saved the third day is 8 (or) 2\u00b3
\nSo,
\nThe pattern for the number of nickels she saves each day will be like
\n2, 2\u00b2, 2\u00b3, …..
\nNow,
\nThe number of days it will take for her to save more than 500 nickels is: 28<\/sup>
\nHence, from the above,
\nWe can conclude that
\nThe number of days she will take to save more than 500 nickels each day is about 8 days<\/p>\n

Assessment Practice<\/strong><\/p>\n

Question 38.
\nSelect all expressions equivalent to 5 \u00d7 5 \u00d7 5 \u00d7 5.
\n\"Envision 51<\/sup> \u00d7 54<\/sup>
\n\"Envision 54<\/sup>
\n\"Envision 52<\/sup> \u00d7 52<\/sup>
\n\"Envision 45<\/sup>
\n\"Envision 4(51<\/sup>)
\nAnswer:
\nThe given expression is: 5 \u00d7 5 \u00d7 5 \u00d7 5
\nNow,
\nThe representation of the given expression in the form of an exponent is:
\n5 \u00d7 5 \u00d7 5 \u00d7 5 = 5\u00b2 \u00d7 5\u00b2
\n= 54<\/sup>
\nHence, from the above,
\nWe can conclude that
\nAll the expressions equivalent to 5 \u00d7 5 \u00d7 5 \u00d7 5 are:
\n\"\"<\/p>\n

Question 39.
\nWhich expression is equivalent to \\(\\frac{1}{36}\\)?
\nA. \\(\\frac{1}{3}\\) \u00d7 \\(\\frac{1}{6}\\)
\nB. \\(\\frac{1}{4}\\) \u00d7 (\\(\\frac{1}{3}\\))3<\/sup>
\nC. (\\(\\frac{1}{2}\\))2<\/sup> \u00d7 (\\(\\frac{1}{3}\\))2<\/sup>
\nD. \\(\\frac{1}{2}\\) \u00d7 \\(\\frac{1}{3}\\) \u00d7 \\(\\frac{1}{3}\\) \u00d7 \\(\\frac{1}{3}\\)
\nAnswer:
\nThe given expression is: \\(\\frac{1}{36}\\)
\nNow,
\nThe representation of the givene xpression in various forms is:
\n\\(\\frac{1}{36}\\) = (\\(\\frac{1}{6}\\))\u00b2
\n= \\(\\frac{1}{9}\\) \u00d7 \\(\\frac{1}{4}\\)
\n= (\\(\\frac{1}{3}\\))\u00b2 \u00d7 (\\(\\frac{1}{2}\\))\u00b2
\nHence, from the above,
\nWe can conclude that
\nThe expression is equivalent to \\(\\frac{1}{36}\\) is:
\n\"\"<\/p>\n

Lesson 3.2 Find Greatest Common Factor and Least Common Multiple<\/h3>\n

Solve & Discuss It!<\/strong><\/p>\n

ACTIVITY<\/strong><\/p>\n

Mark sets the dinner table every 2 days and dries the dishes every 3 days. If he sets the table on Day 2 and dries the dishes on Day 3, on what day would Mark first perform both chores on the same day?
\n\"Envision
\nAnswer:
\nIt is given that
\nMark sets the dinner table every 2 days and dries the dishes every 3 days and he sets the table on Day 2 and dries the dishes on Day 3
\nNow,
\nTo find the days that Mark would perform both chores on the same day,
\nFind the common multiplies of 2 and 3
\nSo,
\nThe common multiples of 2 and 3 are: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60….
\nHence, from the above,
\nWe can conclude that
\nThe days that Mark would perform both chores on the same day must be the multiple of 6<\/p>\n

Look for Relationships<\/strong> What is the relationship between the chores that Mark does each day?
\n\"Envision
\nAnswer:
\nThe completed list of days and chores is:
\n\"\"
\nNow,
\nFrom the above list,
\nWe can observe that
\nFor every day that is multiple of 2,
\nMark sets the dinner table
\nFor every day that is multiple of 3,
\nMark dries the dishes
\nFor every day that is both the multiple of 2 and 3,
\nMark sets the dinner table and dries the dishes
\nHence, from the above,
\nWe can conclude that
\nThe relationship between the chores Mark does each day is:
\na. For every day that is multiple of 2,
\nMark sets the dinner table
\nb. For every day that is multiple of 3,
\nMark dries the dishes
\nc. For every day that is both the multiple of 2 and 3,
\nMark sets the dinner table and dries the dishes<\/p>\n

Focus on math practices<\/strong>
\nGeneralize<\/strong> on what day will Mark do both chores on the same day again? How can you find on which days Mark does both chores without making a list?
\nAnswer:
\nFrom the above problem,
\nWe can observe that
\nThe days that Mark would perform both chores on the same day must be the multiple of 6 i.e., the common multiple of both 2 and 3
\nHence, from the above,
\nWe can conclude that
\nThe days that Mark would perform both chores on the same day must be the multiple of 6 i.e., the common multiple of both 2 and 3<\/p>\n

VISUAL LEARNING<\/strong><\/p>\n

? Essential Question<\/strong> How can you write the prime factorization and find the greatest common factor and least common multiple of two numbers?
\nAnswer:
\nThe steps to Write Prime Factorization of Composite Numbers are:
\na. Put a prime factor on the outside left and the result or quotient (the number of times it divides evenly) underneath.
\nb. Divide the quotient (the number underneath) by another prime number and keep doing this until the bottom number is a prime
\nExamples:
\n\"\"
\nThe steps to find the GCF of a set of numbers, using prime factorization are:
\na. List the prime factors of each number.
\nb. Circle every common prime factor i.e., every prime factor that’s a factor of every number in the set.
\nc. Multiply all the circled numbers. The result is the GCF.
\nThe steps to find the LCM using the prime factorization method are:
\na. Find the prime factorization of each number.
\nb. Write each number as a product of primes, matching primes vertically when possible.
\nc. Bring down the primes in each column.
\nd. Multiply the factors to get the LCM.<\/p>\n

Try It!<\/strong>
\nFind the prime factorization of 56. Start with the least prime factor.
\n\"Envision
\nAnswer:
\nThe given number is: 56
\nNow,
\nBy using the Prime factorization method,
\n\"\"
\nSo,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe prime factorization of 56 is: 2\u00b3 \u00d7 7<\/p>\n

Convince Me!<\/strong> A number is greater than 2 and it has 2 as a factor. Is the number prime or composite? Explain.
\nAnswer:
\nIt is given that
\nA number is greater than 2 and it has 2 as a factor
\nNow,
\nWe know that,
\n2 is both prime and composite. The definition of prime is a number whose two factors are one and itself. If it has two, it must be composite, because that is not one or itself
\nNow,
\nAccording to the given information,
\nOne of the numbers that are greater than 2 and have 2 as a factor will be: 4
\nNow,
\nWe know that,
\n4 has more than 2 factors
\nHence, from the above,
\nWe can conclude that
\nA number that is greater than 2 and has 2 as a factor is a “Composite number”<\/p>\n

Try It!<\/strong>
\nKeesha has 24 beads to add equally to each bag. Can she still make 6 bags and have no supplies left over? Explain.
\nAnswer:
\nIt is given that
\nKeesha has 24 beads to add equally to each bag.
\nNow,
\nThe GCF of 24 and 6 is:
\n\"\"
\nNow,
\nWe can say that
\nThe highest number of bags Keesha can make is: 6 bags
\nHence, from the above,
\nWe can conclude that
\nKeesha can still make 6 bags and have no supplies leftover<\/p>\n

Try It!<\/strong>
\nUse the GCF and the Distributive Property to find the sum of 12 and 36.
\nAnswer:
\nThe given numbers are: 12 and 36
\nNow,
\nStep 1:
\nThe GCF of 12 and 36 is:
\n\"\"
\nStep 2:
\nWrite 12 and 36 as a product using the GCF as a factor
\nNow,
\nBy using the Distributive Property,
\n12 + 36 = (12 \u00d7 1) + (12 \u00d7 3)
\n= 12 \u00d7 (1 + 3)
\n= 12 \u00d7 4
\n= 48
\nHence, from the above,
\nWe can conclude that
\nThe sum of 12 and 36 is: 48<\/p>\n

Try It!<\/strong>
\nGrant also buys bottled water and juice pouches for the picnic. There are 12 bottles of water in each case and 10 juice pouches in each box. Grant wants to buy the least amount but still have as many bottles of water as juice pouches. How many of each should he buy? Explain.
\nAnswer:
\nIt is given that
\nGrant also buys bottled water and juice pouches for the picnic. There are 12 bottles of water in each case and 10 juice pouches in each box. Grant wants to buy the least amount but still have as many bottles of water as juice pouches
\nNow,
\nTo find the number of bottles and juice pouches Grant bought,
\nFind the LCM of 12 and 10
\nNow,
\nThe LCM of 12 and 10 is:
\n\"\"
\nSo,
\nThe number of bottles Grant should buy = \\(\\frac{60}{12}\\)
\n= 5 bottles
\nThe number of juice pouches Grant should buy = \\(\\frac{60}{10}\\)
\n= 6 juice pouches
\nHence, from the above,
\nWe can conclude that
\nThe number of bottles Grant should buy is: 5 bottles
\nThe number of juice pouches Grant should buy is: 6 juice pouches<\/p>\n

KEY CONCEPT<\/strong>
\nThe greatest common factor (GCF) of two numbers is the greatest number that is a factor of both numbers.
\nFactors of 12: 1, 2, 3, 4, 6, 12
\nFactors of 40: 1, 2, 4, 5, 8, 10, 20, 40
\n2 and 4 are common factors of 12 and 40. 4 is the greatest common factor.
\nThe GCF of 12 and 40 is 4.<\/p>\n

The least common multiple (LCM) of two numbers is the least multiple, not including zero, common to both numbers. Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48 …
\nMultiples of 9: 9, 18, 27, 36, 45, 54 …
\n18 and 36 are common multiples of 6 and 9.
\n18 is the least common multiple.
\nThe LCM of 6 and 9 is 18.<\/p>\n

Do You Understand?<\/strong><\/p>\n

Question 1.
\n? Essential Question<\/strong> How can you write the prime factorization and find the greatest common factor and the least common multiple of two numbers?
\nAnswer:
\nThe steps to Write Prime Factorization of Composite Numbers are:
\na. Put a prime factor on the outside left and the result or quotient (the number of times it divides evenly) underneath.
\nb. Divide the quotient (the number underneath) by another prime number and keep doing this until the bottom number is a prime
\nExamples:
\n\"\"
\nThe steps to find the GCF of a set of numbers, using prime factorization are:
\na. List the prime factors of each number.
\nb. Circle every common prime factor i.e., every prime factor that’s a factor of every number in the set.
\nc. Multiply all the circled numbers. The result is the GCF.
\nThe steps to find the LCM using the prime factorization method are:
\na. Find the prime factorization of each number.
\nb. Write each number as a product of primes, matching primes vertically when possible.
\nc. Bring down the primes in each column.
\nd. Multiply the factors to get the LCM.<\/p>\n

Question 2.
\nWhat are two different ways in which you can use prime factorization to find the prime factors of a number?
\nAnswer:
\nThere are two common ways to perform prime factorization. The first is called the “Prime Factor Tree”, and the second is known as the “Upside-Down Division”
\nAn example of the Prime Factor Tree method is:
\n\"\"
\nAn example of the Upside Down Division method is:
\n\"\"<\/p>\n

Question 3.
\nGeneralize<\/strong> Why is the GCF of two prime numbers always 1?
\nAnswer:
\nGCF stands for Greatest Common Factor. Since 1 is a factor of every number, any two or more numbers have a GCF, because even if the numbers are prime, or if they don’t share any prime factors, then the GCF will be 1. If they do share any prime factors, then the GCF will be the product of those shared prime factors.<\/p>\n

Question 4.
\nConstruct Arguments<\/strong> In example 4, Grant finds applesauce that comes in packages of 8, but now he finds juice bottles in only packages of 3. Will the LCM change? Explain.
\nAnswer:
\nIt is given that
\nIn example 4, Grant finds applesauce that comes in packages of 8, but now he finds juice bottles in only packages of 3
\nNow,
\nIn Example 4,
\nWe can observe that
\nGrant should buy 8 packages of juice bottles and 6 packages of applesauce
\nNow,
\nFrom the given information,
\nWe can observe that
\nInstead of 6 packages of applesauce, there are only 3 packages of applesauce now
\nNow,
\nWe know that,
\nSince 3 and 6 are multiples, the LCM will be the same as in Example 4
\nHence, from the above,
\nWe can conclude that
\nThe LCM will not change<\/p>\n

Question 5.
\nCritique Reasoning<\/strong> Sarah says that you can find the LCM of any two whole numbers by multiplying them together. Provide a counterexample to show that Sarah’s statement is incorrect.
\nAnswer:
\nWe know that,
\n“LCM” means the “Lowest Common Multiple”
\nNow,
\nTo find the LCM of the two whole numbers means finding the least common multiple of the two whole numbers
\nExample:
\nFind the LCM of 30 and 40
\nNow,
\n30 \u00d7 40 = 1,200
\nBut,
\n1,200 is not the LCM of 30 and 40
\nNow,
\nThe LCM of 30 and 40 are:
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nSarah’s statement is incorrect<\/p>\n

Do You Know How?<\/strong><\/p>\n

In 6-8, write the prime factorization of each number. If the number is prime, write prime.<\/strong><\/p>\n

Question 6.
\n33
\nAnswer:
\nThe given number is: 33
\nSo,
\nThe representation of the factors for the given number using the prime factorization method is:
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe factors for the given number using the Prime factorization method is:
\n33 = 3 \u00d7 11<\/p>\n

Question 7.
\n32
\nAnswer:
\nThe given number is: 32
\nSo,
\nThe representation of the factors for the given number using the prime factorization method is:
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe factors for the given number using the Prime factorization method is:
\n32 = 2 \u00d7 2 \u00d7 2 \u00d7 2 \u00d7 2<\/p>\n

Question 8.
\n19
\nAnswer:
\nThe given number is: 19
\nNow,
\nWe know that,
\n19 has the factors of 1 and itself
\nHence, from the above,
\nWe can conclude that the given number is a prime number<\/p>\n

In 9-11, find the GCF for each pair of numbers.<\/strong><\/p>\n

Question 9.
\n18, 36
\nAnswer:
\nThe given numbers are: 18, 36
\nNow,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe GCF of 18 and 36 is: 18<\/p>\n

Question 10.
\n22, 55
\nAnswer:
\nThe given numbers are: 22, 55
\nNow,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe GCF of 22 and 55 is: 11<\/p>\n

Question 11.
\n100, 48
\nAnswer:
\nThe given numbers are: 100, 48
\nNow,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe GCF of 100 and 48 is: 4<\/p>\n

In 12-14, find the LCM for each pair of numbers.<\/strong><\/p>\n

Question 12.
\n2, 5
\nAnswer:
\nThe given numbers are: 2, 5
\nNow,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe LCM of 2 and 5 is: 10<\/p>\n

Question 13.
\n8, 12
\nAnswer:
\nThe given numbers are: 8, 12
\nNow,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe LCM of 8 and 12 is: 24<\/p>\n

Question 14.
\n8, 10
\nAnswer:
\nThe given numbers are: 8, 10
\nNow,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe LCM of 8 and 10 is: 40<\/p>\n

Practice & Problem Solving<\/strong><\/p>\n

Leveled Practice<\/strong>
\nIn 15-18, find the prime factorization of each number. If it is prime, write prime.<\/strong><\/p>\n

Question 15.
\n\"Envision
\nAnswer:
\nThe given number is: 27
\nNow,
\nThe representation of the factors for the given number using the Prime factorization method is:
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe factors of 27 by using the Prime Factorization method is: 3 \u00d7 3 \u00d7 3<\/p>\n

Question 16.
\n\"Envision
\nAnswer:
\nThe given number is: 30
\nNow,
\nThe representation of the factors for the given number using the Prime factorization method is:
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe factors of 30 by using the Prime Factorization method is: 2 \u00d7 3 \u00d7 5<\/p>\n

Question 17.
\n26
\nAnswer:
\nThe given number is: 26
\nNow,
\nThe representation of the factors for the given number using the Prime factorization method is:
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe factors of 26 by using the Prime Factorization method is: 2 \u00d7 13<\/p>\n

Question 18.
\n47
\nAnswer:
\nThe given number is: 47
\nNow,
\nWe know that,
\n47 has factors of 1 and only itself
\nHence, from the above,
\nWe can conclude that
\n47 is a prime number<\/p>\n

In 19-21, find the GCF for each pair of numbers.<\/strong><\/p>\n

Question 19.
\n21, 49
\nAnswer:
\nThe given numbers are: 21, 49
\nNow,
\n\"\"
\nHence, from the above,
\nWe can conclude that GCF of 21, 49 is: 7<\/p>\n

Question 20.
\n8, 52
\nAnswer:
\nThe given numbers are: 8, 52
\nNow,
\n\"\"
\nHence, from the above,
\nWe can conclude that GCF of 8, 52 is: 4<\/p>\n

Question 21.
\n32, 81
\nAnswer:
\nThe given numbers are: 32, 81
\nNow,
\n\"\"
\nHence, from the above,
\nWe can conclude that GCF of 32, 81 is: 1<\/p>\n

In 22-24, use the GCF and the Distributive Property to find each sum.<\/strong><\/p>\n

Question 22.
\n30 + 66
\nAnswer:
\nThe given expression is: 30 + 66
\nNow,
\nStep 1: Find the GCF of 30 and 66
\nSo,
\n\"\"
\nSo,
\nThe GCF of 30 and 66 is: 6
\nStep 2: Write each number as a product using GCF as a factor
\nSo,
\n30 + 66 = (6 \u00d7 5) + (6 \u00d7 11)
\n= 6 \u00d7 (5 + 11)
\n= 6 \u00d7 16
\n= 96
\nHence, from the above,
\nWe can conclude that
\nThe sum of 30 and 66 is: 96<\/p>\n

Question 23.
\n34 + 51
\nAnswer:
\nThe given expression is: 34 + 51
\nNow,
\nStep 1: Find the GCF of 34 and 51
\nSo,
\n\"\"
\nSo,
\nThe GCF of 34 and 51 is: 17
\nStep 2: Write each number as a product using GCF as a factor
\nSo,
\n34 + 51 = (17 \u00d7 2) + (17 \u00d7 3)
\n= 17 \u00d7 (2 + 3)
\n= 17 \u00d7 5
\n= 85
\nHence, from the above,
\nWe can conclude that
\nThe sum of 34 and 51 is: 85<\/p>\n

Question 24.
\n15 + 36
\nAnswer:
\nThe given expression is: 15 + 36
\nNow,
\nStep 1: Find the GCF of 15 and 36
\nSo,
\n\"\"
\nSo,
\nThe GCF of 15 and 36 is: 3
\nStep 2: Write each number as a product using GCF as a factor
\nSo,
\n15 + 36 = (3 \u00d7 5) + (3 \u00d7 12)
\n= 3 \u00d7 (5 + 12)
\n= 3 \u00d7 17
\n= 51
\nHence, from the above,
\nWe can conclude that
\nThe sum of 15 and 36 is: 51<\/p>\n

In 25-27, find the LCM for each pair of numbers.<\/strong><\/p>\n

Question 25.
\n12, 11
\nAnswer:
\nThe given numbers are: 12, 11
\nNow,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe LCM of 12 and 11 is: 132<\/p>\n

Question 26.
\n4, 12
\nAnswer:
\nThe given numbers are: 4, 12
\nNow,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe LCM of 4 and 12 is: 12<\/p>\n

Question 27.
\n5, 8
\nAnswer:
\nThe given numbers are: 5, 8
\nNow,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe LCM of 5 and 8 is: 40<\/p>\n

Question 28.
\nCritique Reasoning<\/strong> Gabrielle and John each wrote the prime factorization of 64. Analyze their work and explain any errors.
\n\"Envision
\nAnswer:
\nIt is given that
\nGabrielle and John each wrote the prime factorization of 64
\nNow,
\nAccording to Gabrielle,
\nThe factors of 64 by using the Prime factorization method is:
\n\"\"
\nSo,
\nThe representation of factors of 64 using the Prime factorization method is:
\n64 = 2 \u00d7 2 \u00d7 2 \u00d7 2 \u00d72 \u00d7 2 = 26<\/sup>
\nNow,
\nAccording to John,
\nThe factors of 64 are:
\n64 = 2 \u00d7 32
\n= 2 \u00d7 2 \u00d716
\n= 2 \u00d7 2 \u00d7 2 \u00d7 8
\n= 2 \u00d7 2 \u00d7 2 \u00d7 2 \u00d7 4
\n= 2 \u00d7 2 \u00d7 2 \u00d7 2 \u00d7 2 \u00d7 2
\nHence, from the above,
\nWe can conclude that
\nThere are no errors in Gabriel’s and John’s work<\/p>\n

Question 29.
\nTo celebrate its grand opening, a store is giving customers gift certificates. Which customer is the first to get two gift certificates?
\n\"Envision
\nAnswer:
\nIt is given that
\nTo celebrate its grand opening, a store is giving customers gift certificates.
\nNow,
\nThe given figure is:
\n\"Envision
\nNow,
\nTo find the customer who is the first that will get two gift certificates,
\nFind the LCM of 6 and 8
\nNow,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe 24th customer is the first to get the two gift certificates<\/p>\n

Question 30.
\nA Model with Math<\/strong>
\nThe Venn diagram at the right shows the factors of 24 and 40.
\na. What is the meaning of each of the three shaded regions?
\n\"Envision
\nAnswer:
\nThe given numbers are: 24 and 40
\nNow,
\nThe given Venn diagram is:
\n\"Envision
\nNow,
\nFrom the given Venn diagram,
\nWe can observe that
\nThe left side region of the Venn diagram shows: The factors of 24
\nThe right side region of the Venn diagram shows: The factors of 40
\nThe midle region of the Venn diagram shows: The common factors of 24 and 40
\nHence, from the above,
\nWe can conclude that
\nThe meaning of the shaded regions is:
\nThe left side region of the Venn diagram shows: The factors of 24
\nThe right side region of the Venn diagram shows: The factors of 40
\nThe midle region of the Venn diagram shows: The common factors of 24 and 40<\/p>\n

b. Explain how you use the Venn diagram to find the GCF of 24 and 40. What is the GCF of 24 and 40?
\nAnswer:
\nThe given Venn diagram is:
\n\"Envision
\nNow,
\nWe know that,
\nThe GCF of the two numbers can be find out by identifying the highest factor in the common factors of a and b
\nSo,
\nIn he given Venn diagram,
\nThe middle region represents the common factors of 24 and 40
\nIn the common factors of 24 and 40, 8 is the highest factor
\nHence, from the above,
\nWe can conclude that
\nThe GCF of 24 and 40 is: 8<\/p>\n

Question 31.
\nReasoning<\/strong> You have 50 blueberry scones and 75 cranberry scones. You want to make as many identical bags as possible. Each bag should have an equal number of blueberry scones and an equal number of cranberry scones. What is the greatest number of bags you can fill? Explain.
\nAnswer:
\nIt is given that
\nYou have 50 blueberry scones and 75 cranberry scones. You want to make as many identical bags as possible. Each bag should have an equal number of blueberry scones and an equal number of cranberry scones
\nNow,
\nTo find the greatest number of bags that you can fill an equal number of blueberry scones and cranberry scones,
\nFind the GCF of 50 and 75
\nNow,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe greatest number of blueberry scones bags and cranberry scones bags that you can fill is: 25 bags<\/p>\n

Question 32.
\nMake Sense and Persevere<\/strong> The prime factorizations of A and B are shown. Find the value of n that needs to be listed as a prime factor of B so that the greatest common factor (GCF) of A and B is 9.
\nPrime factorization of A: 3 \u00d7 3 \u00d7 3
\nPrime factorization of B: 2 \u00d7 2 \u00d7 3 \u00d7 n
\nAnswer:
\nIt is given that
\nThe prime factorizations of A and B are shown and the greatest common factor (GCF) of A and B is 9.
\nNow,
\nThe given prime factorizations of A and B are:
\nPrime factorization of A: 3 \u00d7 3 \u00d7 3
\nPrime factorization of B: 2 \u00d7 2 \u00d7 3 \u00d7 n
\nNow,
\nTo make GCF of A and B to be 9,
\nThe numbers must be 27 and 36
\nNow,
\n27 = 3 \u00d7 3 \u00d7 3
\n36 = 2 \u00d7 2 \u00d7 3 \u00d7 3
\nSo,
\nThe missing number is: 3
\nHence, from the above,
\nWe can conclude that
\nThe value of n is: 3<\/p>\n

Question 33.
\nHigher-Order Thinking<\/strong> Gena has 28 trading cards, Sam has 91 trading cards, and Tiffany has 49 trading cards. Use the GCF and the Distributive Property to find the total number of trading cards Gena, Sam, and Tiffany, have.
\nAnswer:
\nIt is given that
\nGena has 28 trading cards, Sam has 91 trading cards, and Tiffany has 49 trading cards.
\nNow,
\nTo find the total number of trading cards Gena, Sam, and Tiffany have, the following steps that used to solve are:
\nStep 1: Find the GCF of 28, 91, and 49
\nSo,
\n\"\"
\nSo,
\nThe GCF of 28, 91, and 49 is: 7
\nStep 2: Write each number as a product using the GCF as a factor
\nSo,
\n28 + 91 + 49 = (7 \u00d7 4) + (7 \u00d7 13) + (7 \u00d7 7)
\n= 7 \u00d7 (4 + 13 + 7)
\n= 7 \u00d7 24
\n= 168 trading cards
\nHence, from the above,
\nWe can conclude that
\nThe total number of trading cards Gena, Sam, and Tiffany have is: 168 trading cards<\/p>\n

Question 34.
\nPeriodical cicada species emerge in large numbers from their larval stage at different yearly intervals. What is the GCF of the years?
\n\"Envision
\nAnswer:
\nIt is given that
\nPeriodical cicada species emerge in large numbers from their larval stage at different yearly intervals.
\nNow,
\nThe given figure is:
\n\"Envision
\nNow,
\nFrom the given figure,
\nWe can observe that
\nThe two years are: 13 years and 17 years
\nNow,
\nThe GCF of 13 and 17 is:
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe GCF of the given years is: 1<\/p>\n

Question 35.
\nPeople are waiting in line for a theater premiere. Every 5th person in line will receive a free theater ticket. Every 6th person will receive a gift card for $40. Which person is the first to receive both prizes?
\nAnswer:
\nIt is given that
\nPeople are waiting in line for a theater premiere. Every 5th person in line will receive a free theater ticket. Every 6th person will receive a gift card for $40
\nNow,
\nTo find the person who is the first to receive both prizes,
\nFind the LCM of 5 and 6
\nNow,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe 30th person is the first to receive both prizes<\/p>\n

Question 36.
\nTwo volunteer groups plant trees. Group A plants the trees in clusters of 3. Group B plants the trees in clusters of 10. Both groups plant the same number of trees. What is the least number of clusters that Group B plants?
\nAnswer:
\nIt is given that
\nTwo volunteer groups plant trees. Group A plants the trees in clusters of 3. Group B plants the trees in clusters of 10. Both groups plant the same number of trees
\nNow,
\nTo find the least number of clusters that Group B plants,
\nFind the LCM of 3 and 10
\nNow,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe least number of clusters that Group B plants is: 30 clusters<\/p>\n

Question 37.
\nFind the LCM of the two numbers. Then use the LCM to find the corresponding letter in the key. Write that letter in the box. What word did you decode?
\n\"Envision
\nAnswer:
\nThe given data is:
\n\"\"
\nNow,
\na.
\nThe LCM of 2 and 3 is:
\n\"\"
\nSo,
\nThe LCM of 2 and 3 is: 6
\nThe letter that corresponds to the LCM of 2 and 3 is: F
\nb.
\nThe LCM of 3 and 7 is:
\n\"\"
\nSo,
\nThe LCM of 3 and 7 is: 21
\nThe letter that corresponds to the LCM of 3 and 7 is: U
\nc.
\nThe LCM of 2 and 7 is:
\n\"\"
\nSo,
\nThe LCM of 2 and 7 is: 14
\nThe letter that corresponds to the LCM of 2 and 7 is: N
\nSo,
\nThe word did you decode is: FUN
\nHence, from the above,
\nWe can conclude that
\n\"\"
\nThe word you decoded is: FUN<\/p>\n

Question 38.
\nRami has swimming lessons every 3 days and guitar lessons every 8 days. If he has both lessons on the first day of the month, in how many days will Rami have both lessons on the same day again?
\nAnswer:
\nIt is given that
\nRami has swimming lessons every 3 days and guitar lessons every 8 days and he has both lessons on the first day of the month
\nNow,
\nTo find the number of days will Rami have both lessons on the same day again,
\nFind the LCM of 3 and 8
\nNow,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nIn 24 days, Rami will have both lessons on the same day again<\/p>\n

Question 39.
\nA number is between 58 and 68. It has prime factors of 2, 3, and 5. What is the number?
\nAnswer:
\nIt is given that
\nA number is between 58 and 68. It has prime factors of 2, 3, and 5
\nNow,
\nLet the number that has the given prime factors be x
\nNow,
\nIf x has the prime factor of 5, then it must be a multiple of 5
\nSo,
\nThe multiples of 5 between 58 and 68 are: 60, 65
\nNow,
\nThe factors of 65 using the Prime Factorization method are:
\n65 = 5 \u00d7 13
\nThe factors of 60 using the Prime Factorization method are:
\n60 = 2 \u00d7 30
\n= 2 \u00d7 2 \u00d7 15
\n= 2 \u00d7 2 \u00d7 3 \u00d7 5
\nSo,
\n60 has the prime factors of 2, 3, and 5 as given
\nHence, from the above,
\nWe can conclude that
\nThe number that is between 58 and 68 and has the prime factors 2, 3, and 5 is: 60<\/p>\n

Question 40.
\nA college offers shuttle service from Dickson Hall or Lot B to its campus quad. Both shuttles first depart their locations at 9:10 A.M. They run from each location to campus and back at the intervals shown. When is the next time both shuttles will depart for the campus at the same time? Explain.
\n\"Envision
\nAnswer:
\nIt is given that
\nA college offers shuttle service from Dickson Hall or Lot B to its campus quad. Both shuttles first depart their locations at 9:10 A.M. They run from each location to campus and back at the intervals shown.
\nNow,
\nThe given figure is:
\n\"Envision
\nNow,
\nFrom the given figure,
\nWe can observe that
\nThe shuttle departs every 10 minutes from Lot B
\nThe shuttle departs every 12 minutes from Dickson Hall
\nNow,
\nTo find the next time both shuttles will depart for the campus at the same time,
\nFind the LCM of 10 and 12
\nNow,
\n\"\"
\nNow,
\nWe know that,
\n1 hour = 60 minutes
\nSo,
\nThe next time both shuttles will depart from the campus at the same time = (The first time both shuttles departed from the campus at the same time) + 1 hour
\n= 9:10 A.M + 1 hour
\n= 10:10 A.M
\nHence, from the above,
\nWe can conclude that
\nThe next time both shuttles will depart from the campus at the same time is: 10:10 A.M<\/p>\n

Assessment Practice<\/strong><\/p>\n

Question 41.
\nMatch each pair of numbers with the pair(s) of numbers that have the same LCM.
\n\"Envision
\nAnswer:
\nThe given pairs of numbers are: (6, 9), (3, 4), (6, 12), (2, 9), (3, 9), (4, 6), and (9, 12)
\nNow,
\nThe LCMs of the given pairs of numbers are:
\nThe LCM of 6 and 9 is: 18
\nThe LCM of 3 and 4 is: 12
\nThe LCM of 6 and 12 is: 12
\nThe LCM of 2 and 9 is: 18
\nThe LCM of 3 and 9 is: 9
\nThe LCM of 4 and 6 is: 12
\nThe LCM of 9 and 12 is: 36
\nHence, from the above,
\nWe can conclude that
\nThe matched pair of numbers with the pair of numbers that has the same LCM is:
\n\"\"<\/p>\n

Question 42.
\nWhich expression is equivalent to 48 + 60?
\nA. 12(4 + 5)
\nB. 12(8 + 5)
\nC. 6(6 + 10)
\nD. 6(8 + 12)
\nAnswer:
\nThe given expression is: 48 + 60
\nNow,
\nStep 1: Find the GCF of 48 and 60
\nSo,
\n\"\"
\nStep 2: Write each number as a product by using the GCF as a factor
\nSo,
\n48 + 60 = (12 \u00d7 4) + (12 \u00d7 5)
\n= 12 \u00d7 (4 + 5)
\nHence, from the above,
\nWe can conclude that
\nThe expression that is equivalent to 48 + 60 is:
\n\"\"<\/p>\n

Lesson 3.3 Write and Evaluate Numerical expressions<\/h3>\n

Solve & Discuss It!<\/strong><\/p>\n

ACTIVITY<\/strong>
\nAn airline company charges additional fees for bags that do not meet the weight and size limits. For one flight, fees were charged for a total of 50 bags that were over the weight limit and 6 oversized bags. Find the total amount of fees collected for that flight.
\n\"Envision
\nLook for Relationships<\/strong> You can use the order of operations to evaluate numerical expressions.
\nAnswer:
\nIt is given that
\nAn airline company charges additional fees for bags that do not meet the weight and size limits. For one flight, fees were charged for a total of 50 bags that were over the weight limit and 6 oversized bags.
\nNow,
\nThe given figure is:
\n\"Envision
\nSo,
\nThe total amount of fees collected for the given flight = (The total number of overweight bags) \u00d7 (The extra fees for each overweight bag) + (The total number of oversized bags) \u00d7 (The extra fees for each oversized bag)
\n= (50 \u00d7 $49) + (6 \u00d7 $75)
\nNow,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe total amount of fees collected for the given flight is: $2900<\/p>\n

Focus on math practices<\/strong>
\nA Model with Math<\/strong>
\nTamara was charged for two bags that were over the weight limit and another bag that was over the size limit. Write and evaluate a numerical expression to find the additional fees Tamara was charged for her bags.
\nAnswer:
\nIt is given that
\nTamara was charged for two bags that were over the weight limit and another bag that was over the size limit.
\nNow,
\nThe given figure is:
\n\"Envision
\nSo,
\nThe total additional fees Tamara charged for her bags = (The total number of overweight bags) \u00d7 (The extra fees for each overweight bag) + (The total number of oversized bags) \u00d7 (The extra fee for each oversized bag)
\n= (2 \u00d7 $49) + (1 \u00d7 $75)
\nNow,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe total additional fees Tamara charged for her bags is: $173<\/p>\n

VISUAL LEARNING<\/strong><\/p>\n

? Essential Question<\/strong> How do you write and evaluate numerical expressions?
\nAnswer:
\nWe write a numerical expression by using numbers and mathematical operators\u00a0which are addition, subtraction, multiplication, and division.
\nExample:
\nThe expression of the mathematical statement “4 added to 2”, will be 2+4.
\nTo evaluate a numerical expression means to find the value of the expression by using the order of operations. To evaluate an expression, we follow the rules of the order of operations in the expression and then simplify the expression using the order of operations.<\/p>\n

Scan for Multimedia<\/strong><\/p>\n

Try It!<\/strong>
\nEvaluate the numerical expression at the right.
\n\"Envision
\nAnswer:
\nThe given numerical expression is: \\(\\frac{1}{8}\\) [6\u00b3 + (48 \u00f7 6)] – 20
\nNow,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe value of the given numerical expression is: 8<\/p>\n

Convince Me!<\/strong> Why is it important to follow the order of operations?
\nAnswer:
\nThe order of operations is a rule that tells you the right order in which to solve different parts of a math problem. Subtraction, multiplication, and division are all examples of operations. The order of operations is important because it guarantees that people can all read and solve a problem in the same way<\/p>\n

Try It!<\/strong>
\nA. Evaluate the numerical expression: 3.22<\/sup> \u2013 [(9 \u00d7 4) + 9] \u00d7 (\\(\\frac{1}{3}\\))2<\/sup>?
\nAnswer:
\nThe given numerical expression is: 3.22<\/sup> \u2013 [(9 \u00d7 4) + 9] \u00d7 (\\(\\frac{1}{3}\\))2<\/sup>
\nNow,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe value of the given numerical expression is: 5.24<\/p>\n

B. Insert grouping symbols so that the numerical expression has a value of 80.
\n6 + 12 \u00d7 (\\(\\frac{2}{3}\\))2<\/sup> \u00d7 3 + 7
\nAnswer:
\nThe given expression is:
\n6 + 12 \u00d7 (\\(\\frac{2}{3}\\))2<\/sup> \u00d7 3 + 7
\nNow,
\nIt is given that
\nThe value of the given expression is: 80
\nNow,
\nCase 1:
\n6 + 12 \u00d7 (\\(\\frac{2}{3}\\))\u00b2 \u00d7 (3 + 7)
\n= 6 + 12 \u00d7 \\(\\frac{4}{9}\\) \u00d7 10
\n= 6 + \\(\\frac{160}{3}\\)
\n= 6 + 53
\n= 59
\nCase 2:
\n(6 + 12) \u00d7 (\\(\\frac{2}{3}\\))\u00b2 \u00d7 (3 + 7)
\n= 18 \u00d7 \\(\\frac{4}{9}\\) \u00d7 10
\n= 2 \u00d7 4 \u00d710
\n= 8 \u00d7 10
\n= 80
\nHence, from the above,
\nWe can conclude that
\nThe complete numerical expression with inserted grouping symbols and has a value of 80 is:
\n(6 + 12) \u00d7 (\\(\\frac{2}{3}\\))\u00b2 \u00d7 (3 + 7)<\/p>\n

KEY CONCEPT<\/strong>
\nThe order of operations is a set of rules used to evaluate expressions that include more than one operation.<\/p>\n

Order of Operations<\/strong><\/p>\n

1 Evaluate inside grouping symbols,
\nsuch as parentheses or brackets.
\n2 Evaluate powers.
\n3 Multiply or divide from left to right.
\n4 Add or subtract from left to right.<\/p>\n

Do You Understand?<\/strong><\/p>\n

Question 1.
\n? Essential Question<\/strong> How do you write and evaluate numerical expressions?
\nAnswer:
\nWe write a numerical expression by using numbers and mathematical operators\u00a0which are addition, subtraction, multiplication, and division.
\nExample:
\nThe expression of the mathematical statement “4 added to 2”, will be 2+4.
\nTo evaluate a numerical expression means to find the value of the expression by using the order of operations. To evaluate an expression, we follow the rules of the order of operations in the expression and then simplify the expression using the order of operations.<\/p>\n

Question 2.
\nMake Sense and Persevere<\/strong> Explain why grouping symbols can change the value of a numerical expression. Then insert grouping symbols to show four different values for the following expression. 80 \u00f7 8 \u00d7 5 + 4
\nAnswer:
\nWe know that,
\n“Grouping symbols” are used in math to show a part of a math expression or equation that must be solved first, before any other calculations are done. The part between the two grouping symbols is treated like one number. Operations inside grouping symbols can be done before any other operations in order to simplify the problem
\nThe position of grouping symbols decide the single value of the numerical expression based on the order of operations
\nNow,
\nThe given expression is: 80 \u00f7 8 \u00d7 5 + 4
\nCase 1:
\nThe given expression with the grouping symbols is: (80 \u00f7 8) \u00d7 5 + 4
\nSo,
\n\"\"
\nCase 2:
\nThe given expression with the grouping symbols is: 80 \u00f7 (8 \u00d7 5) + 4
\nSo,
\n\"\"
\nCase 3:
\nThe given expression with the grouping symbols is: 80 \u00f7 8 \u00d7 (5 + 4)
\nSo,
\n\"\"
\nCase 4:
\nThe given expression with the grouping symbols is: 80 \u00f7 (8 \u00d7 (5 + 4))
\nSo,
\n\"\"<\/p>\n

Question 3.
\nIn the expression (21 – 3) \u00d7 (7 + 2) \u00f7 (12 – 4), what operation should you perform last? Explain.
\nAnswer:
\nThe given expression is:
\n(21 – 3) \u00d7 (7 + 2) \u00f7 (12 – 4)
\nTo decide the order of operation,
\nWe will follow BODMAS RULE
\nWhere,
\nB – Brackets (or) Grouping symbols
\nO – Of
\nD – Division
\nM – Multiplication
\nA – Addition
\nS – Subtraction
\nHence, from the above,
\nWe can conclude that
\nThe operation you should perform last for the given expression is: Multiplication<\/p>\n

Question 4.
\nCritique Reasoning<\/strong> Charles says that 2 \u00d7 3 – 2 is 4, and Seth says that 2 \u00d7 3 – 2 is 2. Who is correct? Explain.
\nAnswer:
\nIt is given that
\nCharles says that 2 \u00d7 3 – 2 is 4, and Seth says that 2 \u00d7 3 – 2 is 2.
\nNow,
\nWe know that,
\nTo decide the order of operation,
\nWe will follow BODMAS RULE
\nWhere,
\nB – Brackets (or) Grouping symbols
\nO – Of
\nD – Division
\nM – Multiplication
\nA – Addition
\nS – Subtraction
\nSo,
\nAccording to Charles:
\nThe expression is: 2 \u00d7 3 – 2
\nSo,
\n(2 \u00d7 3) – 2
\n= 6 – 2
\n= 4
\nAccording to Seth:
\nThe expression is: 2 \u00d7 3 – 2
\nSo,
\n2 \u00d7 (3 – 2)
\n= 2 \u00d7 1
\n= 2
\nHence, from the above,
\nWe can conclude that
\nCharles is correct since he followed the order of operations correctly<\/p>\n

Do You Know How?<\/strong><\/p>\n

In 5-9, evaluate each expression.<\/strong><\/p>\n

Question 5.
\n52<\/sup> + (6.7 – 3.1)
\nAnswer:
\nThe given expression is: 52<\/sup> + (6.7 – 3.1)
\nSo,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: 28.6<\/p>\n

Question 6.
\n(8.2 + 5.3) \u00f7 5
\nAnswer:
\nThe given expression is: (8.2 + 5.3) \u00f7 5
\nSo,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: 2.7<\/p>\n

Question 7.
\n(1.5 – 0.52<\/sup>) \u00f7 [(3 + 2) \u00d7 2]
\nAnswer:
\nThe given expression is: (1.5 – 0.52<\/sup>) \u00f7 [(3 + 2) \u00d7 2]
\nSo,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: 0.125<\/p>\n

Question 8.
\n36.8 \u00f7 [11.5 – (2.5 \u00d7 3)]2<\/sup>
\nAnswer;
\nThe given expression is: 36.8 \u00f7 [11.5 – (2.5 \u00d7 3)]2<\/sup>
\nSo,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: 2.3<\/p>\n

Question 9.
\n6 + 4 \u00d7 5 \u00f7 2 – 8 \u00d7 1.5
\nAnswer:
\nThe given expression is: 6 + 4 \u00d7 5 \u00f7 2 – 8 \u00d7 1.5
\nSo,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: 4<\/p>\n

In 10-12, insert grouping symbols so that the expression has the given value.<\/strong><\/p>\n

Question 10.
\n12 \u00d7 32<\/sup> + 36 Target value: 540
\nAnswer:
\nThe given expression is: 12 \u00d7 32<\/sup> + 36
\nNow,
\nIt is given that
\nThe value of the given expression is: 540
\nNow,
\nCase 1:
\n12 \u00d7 32<\/sup> + 36
\n= (12 \u00d7 32<\/sup> ) + 36
\n= (12 \u00d7 9) + 36
\n= 108 + 36
\n= 144
\nCase 2:
\n12 \u00d7 3\u00b2 + 36
\n= 12 \u00d7 (3\u00b2 + 36)
\n= 12 \u00d7 (9 + 36)
\n= 12 \u00d7 45
\n= 540
\nHence, from the above,
\nWe can conclude that
\nThe representation of the given expression with the grouping values and a target value of 540 is:
\n12 \u00d7 (3\u00b2 + 36)<\/p>\n

Question 11.
\n32 \u00f7 23<\/sup> – 4 Target value: 8
\nAnswer:
\nThe given expression is: 32 \u00f7 23<\/sup> – 4
\nNow,
\nIt is given that
\nThe value of the given expression is: 8
\nNow,
\nCase 1:
\n32 \u00f7 2\u00b3 – 4
\n= (32 \u00f7 2\u00b3) – 4
\n= (32 \u00f7 8) – 4
\n= 4 – 4
\n= 0
\nCase 2:
\n32 \u00f7 2\u00b3 – 4
\n= 32 \u00f7 (2\u00b3\u00a0 – 4)
\n= 32 \u00f7 (8 – 4)
\n= 32 \u00f74
\n= 8
\nHence, from the above,
\nWe can conclude that
\nThe representation of the given expression with the grouping values and a target value of 8 is:
\n32 \u00f7 (2\u00b3 – 4)<\/p>\n

Question 12.
\n2.32<\/sup> + 9 \u00d7 4 \u00f7 2 Target value: 28.58
\nAnswer:
\nThe given expression is: 2.32<\/sup> + 9 \u00d7 4 \u00f7 2
\nNow,
\nIt is given that
\nThe value of the given expression is: 28.58
\nNow,
\nCase 1:
\n2.32<\/sup> + 9 \u00d7 4 \u00f7 2
\n= 2.32<\/sup> + (9 \u00d7 4) \u00f7 2
\n= 2.32<\/sup> + (36) \u00f7 2
\n= 2.32<\/sup> + 18
\n= 23.29
\nCase 2:
\n2.32<\/sup> + 9 \u00d7 4 \u00f7 2
\n= (2.32<\/sup> + 9 \u00d7 4) \u00f7 2
\n= (5.29 + 36)\u00a0\u00f7 2
\n= 20.64
\nCase 3:
\n2.32<\/sup> + 9 \u00d7 4 \u00f7 2
\n= (2.32<\/sup> + 9) \u00d7 (4 \u00f7 2)
\n= 14.29 \u00d7 2
\n= 28.58
\nHence, from the above,
\nWe can conclude that
\nThe representation of the given expression with the grouping values and a target value of 28.58 is:
\n(2.32<\/sup> + 9) \u00d7 (4 \u00f7 2)<\/p>\n

Practice & Problem Solving<\/strong>
\nLeveled Practice<\/strong>
\nIn 13-18, use the order of operations to evaluate.<\/strong><\/p>\n

Question 13.
\n\"Envision
\nAnswer:
\nThe given expression is: 4\u00b2 – (3.1 + 6.4) + 4.5
\nSo,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: 11<\/p>\n

Question 14.
\n\"Envision
\nAnswer:
\nThe given expression is: (8.7 + 3.3) \u00d7 (\\(\\frac{1}{2}\\))\u00b2
\nSo,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: 3<\/p>\n

Question 15.
\n\"Envision
\nAnswer:
\nThe given expression is: 157.8 – (3\u00b2 + 6) \u00d7 3
\nSo,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: 112.8<\/p>\n

Question 16.
\n4.3 + (8.4 – 5.1)
\nAnswer:
\nThe given expression is: 4.3 + (8.4 – 5.1)
\nSo,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: 7.6<\/p>\n

Question 17.
\n1.25 \u00d7 4 + 3 \u00d7 2 \u00f7 (\\(\\frac{1}{2}\\))3<\/sup>
\nAnswer:
\nThe given expression is: 1.25 \u00d7 4 + 3 \u00d7 2 \u00f7 (\\(\\frac{1}{2}\\))3<\/sup>
\nSo,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: 29<\/p>\n

Question 18.
\n[23<\/sup> \u00d7 (152 \u00f7 8)] – 52
\nAnswer:
\nThe given expression is: [23<\/sup> \u00d7 (152 \u00f7 8)] – 52
\nSo,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: 100<\/p>\n

In 19-21, insert grouping symbols so that the expression has the given value.<\/strong><\/p>\n

Question 19.
\nTarget value: 32
\n2 \u00d7 9 + 7
\nAnswer:
\nThe given expression is: 2 \u00d7 9 + 7
\nNow,
\nIt is given that
\nThe value of the given expression is: 32
\nNow,
\nCase 1:
\n2 \u00d7 9 + 7
\n= (2 \u00d7 9) + 7
\n= 18 + 7
\n= 25
\nCase 2:
\n2 \u00d7 9 + 7
\n= 2 \u00d7 (9 + 7)
\n= 2 \u00d7 16
\n= 32
\nHence, from the above,
\nWe can conclude that
\nThe representation of the given expression with the grouping values and a target value of 32 is:
\n2 \u00d7 (9 + 7)<\/p>\n

Question 20.
\nTarget value: 6
\n\\(\\frac{1}{3}\\) \u00d7 21 – 3
\nAnswer:
\nThe given expression is: \\(\\frac{1}{3}\\) \u00d7 21 – 3
\nNow,
\nIt is given that
\nThe value of the given expression is: 6
\nNow,
\nCase 1:
\n\\(\\frac{1}{3}\\) \u00d7 21 – 3
\n= (\\(\\frac{1}{3}\\) \u00d7 21) – 3
\n= 7 – 3
\n= 4
\nCase 2:
\n\\(\\frac{1}{3}\\) \u00d7 21 – 3
\n= \\(\\frac{1}{3}\\) \u00d7 (21 – 3)
\n= \\(\\frac{1}{3}\\) \u00d7 18
\n= 6
\nHence, from the above,
\nWe can conclude that
\nThe representation of the given expression with the grouping values and a target value of 6 is:
\n\\(\\frac{1}{3}\\) \u00d7 (21 – 3)<\/p>\n

Question 21.
\nTarget value: 43
\n2.5 + 5 \u00d7 6 – 2
\nAnswer:
\nThe given expression is: 2.5 + 5 \u00d7 6 – 2
\nNow,
\nIt is given that
\nThe value of the given expression is: 43
\nNow,
\nCase 1:
\n2.5 + 5 \u00d7 6 – 2
\n= (2.5 + 5) \u00d7 (6 – 2)
\n= 7.5 \u00d7 4
\n= 30
\nCase 2:
\n2.5 + 5 \u00d7 6 – 2
\n= [(2.5 + 5) \u00d7 6] – 2
\n= (7.5 \u00d7 6) – 2
\n= 45 – 2
\n= 43
\nHence, from the above,
\nWe can conclude that
\nThe representation of the given expression with the grouping values and a target value of 43 is:
\n[(2.5 + 5) \u00d7 6] – 2<\/p>\n

Question 22.
\nCory bought some baseball equipment. He used a coupon for \\(\\frac{1}{2}\\) off the price of the bat and glove. Write and evaluate a numerical expression to find the total cost of the bat, the glove, and
\n\"Envision
\nAnswer:
\nIt is given that
\nCory bought some baseball equipment. He used a coupon for \\(\\frac{1}{2}\\) off the price of the bat and glove
\nNow,
\nThe given figure is:
\n\"Envision
\nNow,
\nFrom the given information,
\nThe new cost of the bat will become \\(\\frac{1}{2}\\) of the old price of the bat
\nThe new cost of the glove will become \\(\\frac{1}{2}\\) of the old price of the glove
\nSo,
\nThe total cost of the bat, the glove and 3 baseballs = (The new cost of the bat) + (The new cost of the glove) + (3 \u00d7 The cost of the baseball)
\n= \\(\\frac{$69}{2}\\) + \\(\\frac{$75}{2}\\) + (3 \u00d7 $5.50)
\n= \\(\\frac{1}{2}\\) (69 + 75) + (3 \u00d7 $5.50)
\n= \\(\\frac{$144}{2}\\) + $16.50
\n= $72 + $16.50
\n= $88.50
\nHence, from the above,
\nWe can conclude that
\nThe total cost of the bat, the glove, and 3 baseballs is: $88.50<\/p>\n

Question 23.
\nMake Sense and Persevere<\/strong> Write a numerical expression, with at least three operations, that has the same value as the following expression. Justify your answer.
\n5 + (8 – 4) \u00f7 2 + 3
\nAnswer:
\nThe given expression is: 5 + (8 – 4) \u00f7 2 + 3
\nNow,
\n\"\"
\nSo,
\nThe value of the given expression is: 10
\nNow,
\nThe expression that has the same value as the given expression and has at least 3 operations is:
\n(5 \u00d7 2) + 4 – (36 \u00f7 9)<\/p>\n

Question 24.
\nUse Structure<\/strong> How do you know which part of the numerical expression to evaluate first? Explain.
\n(26 + 2.5) \u2013 [(8.3 \u00d7 3) + (13<\/sup> – 0.25)]
\nAnswer:
\nThe given expression is:
\n(26 + 2.5) \u2013 [(8.3 \u00d7 3) + (13<\/sup> – 0.25)]
\nNow,
\nTo solve an expression,
\nWe will use the order of operations using the BODMAS rule and solve the part of the expression that involves more operations in it
\nSo,
\nIn the given expression,
\nWe have to first evaluate [(8.3 \u00d7 3) + (13<\/sup> – 0.25)]
\nHence, from the above,
\nWe can conclude that
\nBy using the order of operations using the BODMAS rule and the part of the expression that involves more operations in it,
\nWe have to first evaluate [(8.3 \u00d7 3) + (13<\/sup> – 0.25)]<\/p>\n

Question 25.
\nConstruct Arguments<\/strong> Evan says that the value of the numerical expression 0.22<\/sup> + 12 \u00f7 (1.5 \u00d7 4) is 32.04. Do you agree? Explain.
\nAnswer:
\nThe given expression is: 0.22<\/sup> + 12 \u00f7 (1.5 \u00d7 4)
\nSo,
\n\"\"
\nBut,
\nFor the given expression,
\nIt is given that the value is: 32.04
\nBut,
\nWe got the value of the expression as 2.04
\nHence, from the above,
\nWe can conclude that
\nWe can not agree with Evan<\/p>\n

Question 26.
\nThe width of the rectangular drawing is one-third the length plus 3 inches. What is the perimeter of the drawing? Write and evaluate an expression to solve the problem.
\n\"Envision
\nAnswer:
\nIt is given that
\nThe width of the rectangular drawing is one-third the length plus 3 inches.
\nNow,
\nThe given figure is:
\n\"Envision
\nNow,
\nFrom the given figure,
\nWe can observe that
\nThe length of the rectangular drawing is 12 in.
\nNow,
\nAccording to the given information,
\nThe width of the rectangular drawing = (The length of the rectangular drawing \u00f7 3) + 3
\n= \\(\\frac{12}{3}\\) + 3
\n= 4 + 3
\n= 7 in.
\nNow,
\nWe know that,
\nThe perimeter of a rectangle = 2 \u00d7 (Length + Width)
\nSo,
\nThe perimeter of a rectangular drawing = 2 \u00d7 (12 + 7)
\n= 2 \u00d7 19
\n= 38 in.
\nHence, from the above,
\nWe can conclude that
\nThe perimeter of the rectangular drawing is 38 in.<\/p>\n

Question 27.
\nHigher-Order Thinking<\/strong> Frederick evaluates the numerical expression [(53.7 + 37.2) – (33<\/sup> + 3.8)] \u2013 8.6 and records the answer as 51.5. Lana evaluates the numerical expression 53.7 + 37.2 – 33<\/sup> + 3.8 – 8.6 and records the answer as 59.1. The expressions have the same numbers and operations. Explain how Frederick and Lana can both be correct.
\nAnswer:
\nIt is given that
\nFrederick evaluates the numerical expression [(53.7 + 37.2) – (33<\/sup> + 3.8)] \u2013 8.6 and records the answer as 51.5. Lana evaluates the numerical expression 53.7 + 37.2 – 33<\/sup> + 3.8 – 8.6 and records the answer as 59.1. The expressions have the same numbers and operations
\nNow,
\nAccording to Fredrick:
\nThe given expression is: [(53.7 + 37.2) – (33<\/sup> + 3.8)] \u2013 8.6
\nSo,
\n\"\"
\nNow,
\nAccording to Lana:
\nThe given expression is: 53.7 + 37.2 – 33<\/sup> + 3.8 – 8.6
\nSo,
\n\"\"
\nNow,
\nFrom both answers,
\nWe can observe that
\nEven though the operations and the numbers are the same, the only difference between the expressions is the grouping symbols
\nHence, from the above,
\nWe can conclude that
\nBoth Fredrick and Lana are correct and the only difference between their expressions is the grouping of symbols<\/p>\n

Question 28.
\nModel with Math<\/strong>
\nLillian went to the gift shop on the boardwalk and bought four bags of dyed seashells at $3.99 each. She had a coupon for $1 off. Her mom paid for half of the remaining cost. Write and evaluate a numerical expression to find how much Lillian paid toward the purchase of the seashells.
\nAnswer:
\nIt is given that
\nLillian went to the gift shop on the boardwalk and bought four bags of dyed seashells at $3.99 each. She had a coupon for $1 off. Her mom paid for half of the remaining cost.
\nNow,
\nThe cost of 4 bags of dyed seashells = 4 \u00d7 (The cost of each bag of dyed seashells)
\n= 4 \u00d7 $3.99
\n= $15.96
\nNow,
\nIt is given that
\nShe had a coupon for $1 off
\nSo,
\nThe amount Lilian paid to purchase 4 bags of dyed seashells = $15.96 – $1
\n= $14.96
\nNow,
\nIt is given that
\nLilian’s mom paid for half of the remaining cost
\nSo,
\nThe amount Lilian’s mom paid = \\(\\frac{$14.96}{2}\\)
\n= $7.48
\nSo,
\nThe\u00a0 total amount Lilian paid toward the purchase of the seashells = $14.96 – $7.48
\n= $7.48
\nHence, from the above,
\nWe can conclude that
\nThe total amount Lilian paid toward the purchase of the seashells is: $7.48<\/p>\n

Question 29.
\nIn an ecosystem, some animals get energy by eating plants. Write and evaluate an expression to find how many pounds of plants a herd of 18 elk can eat in one week.
\n\"Envision
\nAnswer:
\nIt is given that
\nIn an ecosystem, some animals get energy by eating plants
\nNow,
\nThe given figure is:
\n\"Envision
\nNow,
\nWe know that,
\n1 week = 7 days
\nSo,
\nThe number of pounds of plants a herd of 18 elk can eat in one week = (The number of elks) \u00d7 (The number of pounds of plants an elk eats each day) \u00d7 (The number of days in a\u00a0 week)
\n= (18 \u00d7 20 \u00d7 7) pounds
\n= (126 \u00d7 20) pounds
\n= 2,520 pounds
\nHence, from the above,
\nWe can conclude that
\nThe number of pounds of plants a herd of 18 elk can eat in one week is: 2,520 pounds<\/p>\n

Assessment Practice<\/strong><\/p>\n

Question 30.
\nSelect all expressions that are equivalent to 24<\/sup> \u00f7 [(3.2 \u00d7 0.8) + 1.44].
\n\"Envision 24<\/sup> \u00f7 [(4 \u00d7 0.64) + 1.44]
\n\"Envision 16 \u00f7 [(22<\/sup> \u00d7 0.64) + (0.72 \u00d7 2)]
\n\"Envision 8 \u00f7 [(3.2 \u00d7 0.8) + (0.48 \u00d7 3)]
\n\"Envision 24<\/sup> \u00f7 [2.56 + (0.48 \u00d7 4)]
\n\"Envision 42 \u00f7 [2.56 + (0.48 \u00d7 4)]
\nAnswer:
\nThe given expression is: 24<\/sup> \u00f7 [(3.2 \u00d7 0.8) + 1.44]
\nSo,
\n\"\"
\nNow,
\nThe given expressions and its values are:
\nA.\u00a0 24<\/sup> \u00f7 [(4 \u00d7 0.64) + 1.44] = 4
\nB.\u00a0 16 \u00f7 [(22<\/sup> \u00d7 0.64) + (0.72 \u00d7 2)] = 4
\nC.\u00a0 8 \u00f7 [(3.2 \u00d7 0.8) + (0.48 \u00d7 3)] = 2
\nD.\u00a0 24<\/sup> \u00f7 [2.56 + (0.48 \u00d7 4)] = 3.57
\nE.\u00a0 4\u00b2 \u00f7 [2.56 + (0.48 \u00d7 4)] = 3.57
\nHence, from the above,
\nWe can conclude that
\nAll expressions that are equivalent to 24<\/sup> \u00f7 [(3.2 \u00d7 0.8) + 1.44] are:
\n\"\"<\/p>\n

Question 31.
\nWhich value is equivalent to the expression 18.9 \u00d7 [(2 \u00d7 2.7) \u2013 4.6] \u2013 22<\/sup>?
\nA. 1,112
\nB. 111.2
\nC. 11.12
\nD. 1.112
\nAnswer:
\nThe given expression is: 18.9 \u00d7 [(2 \u00d7 2.7) \u2013 4.6] \u2013 22<\/sup>
\nSo,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe value is equivalent to the expression 18.9 \u00d7 [(2 \u00d7 2.7) \u2013 4.6] \u2013 2\u00b2 is:
\n\"\"<\/p>\n

Topic 3 MID-TOPIC CHECKPOINT<\/h3>\n

Question 1.
\nVocabulary<\/strong> Describe the relationship between the base and the exponent in 43<\/sup>. Lesson 3-1
\nAnswer:
\nThe given expression is: 4\u00b3
\nNow,
\nFrom the given expression,
\nWe can observe that
\n“4” is called the “Base”
\n“3” is called the “Exponent”
\nNow,
\nWe know that,
\nThe number that is repeatedly multiplied is the “Base”
\nThe “Exponent” tells how many times the Base is used as a factor
\nHence, from the above,
\nWe can conclude that
\n“4” is the number that is repeatedly multiplied
\n“3” is the number that tells how many times “4” is used as a factor<\/p>\n

Question 2.
\nWhat is the GCF of 14 and 42? Lesson 3.2
\nAnswer:
\nThe given numbers are: 14 and 42
\nNow,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe GCF of 14 and 42 is: 14<\/p>\n

Question 3.
\nWhich pair of numbers has a GCF of 5? Lesson 3.2
\nA. 15 and 30
\nB. 5 and 21
\nC. 45 and 9
\nD. 20 and 55
\nAnswer:
\nThe given numbers are:
\nA) 15 and 30\u00a0 \u00a0– GCF is 15
\nB) 5 and 21\u00a0 \u00a0 \u00a0– GCF is 1
\nC) 45 and 9\u00a0 \u00a0 \u00a0– GCF is 9
\nD) 20 and 55\u00a0 \u00a0– GCF is 5
\nHence, from the above,
\nWe can conclude that
\nThe pair of numbers that has a GCF of 5 is:
\n\"\"<\/p>\n

Question 4.
\nWhat is the LCM of 12 and 9? Lesson 3.2
\nAnswer:
\nThe given numbers are: 12 and 9
\nSo,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe LCM of 12 and 9 is: 36<\/p>\n

Question 5.
\nEvaluate the numerical expression. Lesson 3.3
\n0.52<\/sup> \u00d7 (20 – 22<\/sup> \u00d7 3) \u00d7 (\\(\\frac{2}{5}\\) \u00d7 25)
\nAnswer:
\nThe given numerical expression is:
\n0.52<\/sup> \u00d7 (20 – 22<\/sup> \u00d7 3) \u00d7 (\\(\\frac{2}{5}\\) \u00d7 25)
\nSo,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe value of the given numerical expression is: 20<\/p>\n

Question 6.
\nSelect all the expressions that are equal to (\\(\\frac{2}{3}\\))2<\/sup>? Lesson 3.1
\n\"Envision \\(\\frac{4}{9}\\)
\n\"Envision \\(\\frac{4}{3}\\)
\n\"Envision \\(\\frac{1}{3}\\) \u00d7 \\(\\frac{1}{3}\\)
\n\"Envision \\(\\frac{1}{9}\\) \u00d7 4
\n\"Envision \\(\\frac{2}{3}\\) \u00d7 \\(\\frac{2}{3}\\)
\nAnswer:
\nThe given expression is: (\\(\\frac{2}{3}\\))\u00b2
\nSo,
\n(\\(\\frac{2}{3}\\))\u00b2
\n= \\(\\frac{2}{3}\\) \u00d7 \\(\\frac{2}{3}\\)
\n= \\(\\frac{2 \u00d7 2}{3 \u00d7 3}\\)
\n= \\(\\frac{4}{9}\\)
\n= 4 \u00d7 \\(\\frac{1}{9}\\)
\nHence, from the baove,
\nWe can conclude that
\nAll the expressions that are equivalent to (\\(\\frac{2}{3}\\))\u00b2 are:
\n\"\"<\/p>\n

Question 7.
\nLiam bought 2 vintage movie posters, 2 rock posters, and 1 rap poster. He applied a $35 gift card to the total purchase and a \\(\\frac{1}{2}\\)-off coupon to the rap poster. Write and evaluate a numerical expression to show how much Liam paid for the posters. Lesson 3.3
\n\"Envision
\nAnswer:
\nIt is given that
\nLiam bought 2 vintage movie posters, 2 rock posters, and 1 rap poster. He applied a $35 gift card to the total purchase and a \\(\\frac{1}{2}\\)-off coupon to the rap poster
\nNow,
\nThe given table is:
\n\"Envision
\nNow,
\nThe amount Liam has to pay for the posters = (The number of vintage movie posters) \u00d7 (The cost of each vintage movie poster) + (The number of rap posters) \u00d7 (The cost of each rap poster) + (The number of rock posters) \u00d7 (The cost of each rock poster)
\n= (2 \u00d7 $28.50) + (1 \u00d7 $19.50) + (2 \u00d7 $29.75)
\nNow,
\n\"\"
\nSo,
\nThe amount Liam has to pay for the posters is: $136
\nNow,
\nWith a $35 gift coupon,
\nThe amount Liam has to pay for the posters = $136 – $35
\n= $101
\nNow,
\nWith the \\(\\frac{1}{2}\\) off coupon of rap poster,
\nThe total cost of rap poster = \\(\\frac{$19.50}{2}\\)
\n= $9.75
\nSo,
\nThe amount Liam has to pay for the posters = $101 – $9.75
\n= $91.25
\nHence, from the above,
\nWe can conclude that
\nThe amount of money Liam paid for the products is: $91.25<\/p>\n

Question 8.
\nEva counts up by 3s, while Jin counts up by 5s. What is the smallest number that they both say? Lesson 3-2
\nAnswer:
\nIt is given that
\nEva counts up by 3s, while Jin counts up by 5s
\nNow,
\nTo find the smallest number that both Evan and Jin can say,
\nFind the LCM of 3 and 5
\nNow,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe smallest number that both Eva and Jin can say is: 15<\/p>\n

How well did you do on the mid-topic checkpoint? Fill in the stars. \"Envision<\/p>\n

Topic 3 MID-TOPIC PERFORMANCE TASK<\/strong><\/p>\n

Monique and Raoul are helping teachers make gift bags and gather supplies for a student celebration day at Pineville Middle School.<\/p>\n

PART A<\/strong>
\nRaoul has 72 wristbands and 96 movie passes to put in gift bags. The greatest common factor for the number of wristbands and the number of movie passes is equal to the number of gift bags Raoul needs to make. Find the number of gift bags Raoul needs to make. Then find how many wristbands and how many movies passes Raoul can put in each gift bag if he evenly distributes the items.
\nAnswer:
\nIt is given that
\nMonique and Raoul are helping teachers make gift bags and gather supplies for a student celebration day at Pineville Middle School.
\nRaoul has 72 wristbands and 96 movie passes to put in gift bags. The greatest common factor for the number of wristbands and the number of movie passes is equal to the number of gift bags Raoul needs to make.
\nNow,
\nAccording to the given information,
\nThe number of gift bags Raoul needs to make = The GCF of the number of wristbands and the number of movies passes
\nSo,
\n\"\"
\nSo,
\nThe number of gift bags Raoul needs to make is: 24 gift bags
\nNow,
\nThe number of wristbands = (The number of gift bags Raoul needs to make) \u00f7 2
\n= \\(\\frac{24}{2}\\)
\n= 12 wristbands
\nThe number of movie tickets = (The number of gift bags Raoul needs to make) \u00f7 2
\n= \\(\\frac{24}{2}\\)
\n= 12 movie tickets
\nHence, from the above,
\nWe can conclude that
\nThe number of gift bags Raoul needs to make is: 24 gift bags
\nThe number of wristbands is: 12
\nThe number of movie tickets is: 12<\/p>\n

PART B<\/strong>
\nMonique wants to have an equal number of cups and napkins. What are the least number of packages of cups and the least number of packages of napkins Monique should buy to have an equal number of cups and napkins? Justify your answer.
\n\"Envision
\nAnswer:
\nIt is given that
\nMonique wants to have an equal number of cups and napkins
\nNow,
\nThe given table is:
\n\"Envision
\nNow,
\nTo find the least number of packages of cups and the least number of packages of napkins,
\nFind the LCM of the number of cups and napkins present in each package
\nSo,
\nWe have to find the LCM of 12 and 10 to find the least number of packages of cups and napkins
\nNow,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe least number of packages of cups and the least number of packages of napkins Monique should buy to have an equal number of cups and napkins is: 60<\/p>\n

PART C<\/strong>
\nWhich numerical expressions show the equal number of both cups and napkins that Monique will have in Part B? Select all that apply.
\n\"Envision 21<\/sup> \u00d7 30
\n\"Envision 102<\/sup> \u00d7 60
\n\"Envision 22<\/sup> \u00d7 15
\n\"Envision 4600<\/sup>
\n\"Envision 40<\/sup> \u00d7 60
\nAnswer:
\nFrom part B,
\nWe can observe that
\nThe least number of packages of cups and the least number of packages of napkins Monique should buy to have an equal number of cups and napkins is: 60
\nHence, from the above,
\nWe can conclude that
\nThe numerical expressions that show the equal number of both cups and napkins that Monique will have in Part B are:
\n\"\"<\/p>\n

PART D<\/strong>
\nThe teachers have $25 to buy supplies. Write and evaluate a numerical expression to show how much more money they will need to buy the cups and napkins.
\nAnswer:
\nIt is given that
\nThe teachers have $25 to buy supplies
\nNow,
\nThe given table is:
\n\"Envision
\nNow,
\nFrom the given table,
\nWe can observe that
\nThe total cost of cups and Napkins = (The number of cups per package) \u00d7 (The cost of each cup) + (The number of Napkins per package) \u00d7 (The cost of each Napkin)
\n= (12 \u00d7 $3.80) + (10 \u00d7 $4.25)
\n= $45.60 + $42.50
\n= $88.10
\nNow,
\nThe amount of more money than the teachers need to buy cups and napkins = (The total cost of cups and Napkins) – (The amount of money the teachers have)
\n= $88.10 – $25
\n= $63.10
\nHence, from the above,
\nWe can conclude that
\nThe more money the teachers will need to buy the cups and napkins is: $63.10<\/p>\n

Lesson 3.4 Write Algebraic expressions<\/h3>\n

Explore It!<\/strong>
\nThe table shows the number of games the Hornets won and the number of games the Lynx won.
\n\"Envision<\/p>\n

A. What pattern do you see in the data in the table? Explain how the pattern relates to the number of games won.
\nAnswer:
\nIt is given that
\nThe table shows the number of games the Hornets won and the number of games the Lynx won.
\nNow,
\nThe given table is:
\n\"Envision
\nNow,
\nFrom the given table,
\nWe can observe that
\nThe number of games won by Lynx is 2 more than the number of games won by Hornets
\nHence, from the above,
\nWe can conclude that
\nThe pattern that is related to the number of games won is:
\nThe number of games won by Lynx = (The number of games won by Hornet) + 2<\/p>\n

B. Look for Relationships<\/strong> Write numerical expressions to relate the number of games won by the Lynx to the number of games won by the Hornets.
\n\"Envision
\nAnswer:
\nFrom part A,
\nWe know that,
\nThe number of games won by Lynx = (The number of games won by Hornet) + 2
\nHence, from the above,
\nWe can conclude that
\nThe numerical expressions that relate the number of games won by the Lynx to the number of games won by the Hornets is:
\n\"\"<\/p>\n

C. Explain how to complete the table above for the Lynx if the Hornets won n games.
\nAnswer:
\nFrom Part A,
\nWe know that,
\nThe number of games won by Lynx = (The number of games won by Hornet) + 2
\nNow,
\nIt is given that
\nHornets won n games
\nHence, from the above,
\nWe can conclude that
\nThe number of games won by Lynx = n + 2
\nWhere,
\nn is the number of games won by Hornets<\/p>\n

Focus on math practices<\/strong>
\nReasoning<\/strong>
\nSuppose the Lynx won g games. What mathematical expression could you write to show how many games the Hornets won? How is this expression related to the expression you wrote to find the number of games the Lynx won when the Hornets won n games? Explain.
\nAnswer:
\nFrom Part A,
\nWe know that,
\nThe number of games won by Lynx = (The number of games won by Hornets) + 2
\nNow,
\nIf Lynx won g games, then
\nThe number of games won by Hornets = g – 2
\nWhere,
\ng is the number of games won by Lynx
\nThe expression when Hornets won n games is the same as the above expression as we replace the variable g with the variable n<\/p>\n

VISUAL LEARNING<\/strong><\/p>\n

? Essential Question<\/strong> How can you write an algebraic expression?
\nAnswer:
\nAlgebraic expressions are useful because they represent the value of an expression for all of the values a variable can take on. Similarly, when we describe an expression in words that includes a variable, we are describing an algebraic expression, an expression with a variable<\/p>\n

Try It!<\/strong>
\nDarius’s sister Rachel bought m mystery books for $6.50 each. Show three ways to write an algebraic expression that represents the total cost of the mystery books.
\nAnswer:
\nIt is given that
\nDarius’s sister Rachel bought m mystery books for $6.50 each
\nSo,
\nThe cost of m books bought by Rachel = (The total number of books) \u00d7 (The cost of each book)
\n= m \u00d7 $6.50
\n= $6.50m
\nNow,
\nThe representation of the algebraic expression that represents the total cost of the mystery books in 3 ways is:
\n$6.50m (or) $6.50 \u00d7 m (or) $6050(m)
\nHence, from the above,
\nWe can conclude that
\nThe representation of the algebraic expression that represents the total cost of the mystery books in 3 ways is:
\n$6.50m (or) $6.50 \u00d7 m (or) $6050(m)<\/p>\n

Convince Me!<\/strong> How do you know that the expressions you wrote for the cost of the mystery books are algebraic expressions?
\nAnswer:
\nWe know that,
\nAlgebraic expressions are\u00a0combinations of variables, numbers, and at least one arithmetic operation.
\nFor example,
\n2x+4y\u22129 is an algebraic expression.
\nNow,
\nWhen we observe the total cost of m mystery books,
\nWe can observe that
\nm is a variable
\nHence, from the above,”
\nWe can conclude that
\nSince the expression that represents the total cost of the mystery books,
\nWe can say that the expression that represents the total cost of the mystery books is an algebraic expression<\/p>\n

Try It!<\/strong><\/p>\n

Write an algebraic expression that represents “8 minus the quantity b divided by 6.”
\nAnswer:
\nThe given algebraic expression in words is: 8 minus the quantity divided by 6
\nSo,
\nThe representation of the given algebraic expression that is in words into a symbolic form is:
\n(8 – b) \u00f7 6 (or) \\(\\frac{8 – b}{6}\\)
\nHence, from the above,
\nWe can conclude that
\nThe representation of the given algebraic expression that is in words into a symbolic form is:
\n(8 – b) \u00f7 6 (or) \\(\\frac{8 – b}{6}\\)<\/p>\n

Try It!<\/strong>
\nHow many terms does the expression r \u00f7 9 + 5.5 have? Explain.
\nAnswer:
\nThe given expression is: r \u00f7 9 + 5.5
\nNow,
\nWe know that,
\nEach part of an expression that is separated by a plus sign or a minus sign is called a “Term”
\nSo,
\nThe number of terms the given expression contains is: 2 terms
\nHence, from the above,
\nWe can conclude that
\nThe number of terms the given expression contains is: 2 terms<\/p>\n

KEY CONCEPT<\/strong>
\nA variable, written as a letter, represents a quantity that can change. You can use a variable to write an algebraic expression that has at least one operation.
\n\"Envision<\/p>\n

Do You Understand?<\/strong><\/p>\n

Question 1.
\n? Essential Question<\/strong> How can you write an algebraic expression?
\nAnswer;
\nAlgebraic expressions are useful because they represent the value of an expression for all of the values a variable can take on. Similarly, when we describe an expression in words that includes a variable, we are describing an algebraic expression, an expression with a variable<\/p>\n

Question 2.
\nBe Precise<\/strong> Identify the variable and the operation in the algebraic expression \\(\\frac{6}{x}\\)
\nAnswer:
\nThe given algebraic expression is: \\(\\frac{6}{x}\\)
\nNow,
\nWe know that,
\nA “Variable”, written as a letter, represents a quantity that can change
\nSo,
\nIn the given expression,
\nThe variable is: x
\nThe operation in the given algebraic expression is: Division
\nHence, from the above,
\nWe can conclude that
\nIn the given expression,
\nThe variable is: x
\nThe operation in the given algebraic expression is: Division<\/p>\n

Question 3.
\nVocabulary<\/strong> Explain why 15 + \\(\\frac{1}{2}\\)n is an algebraic expression.
\nAnswer:
\nThe given expression is: 15 + \\(\\frac{1}{2}\\)n
\nNow,
\nWe know that,
\nAlgebraic expressions are useful because they represent the value of an expression for all of the values a variable can take on. Similarly, when we describe an expression in words that includes a variable, we are describing an algebraic expression, an expression with a variable
\nHence, from the above,
\nWe can conclude that
\n15 + \\(\\frac{1}{2}\\)n is an algebraic expression<\/p>\n

Question 4.
\nReasoning<\/strong> Could you describe the expression 2(3 + 4) as a product of two factors? Explain.
\nAnswer:
\nThe given expression is: 2 (3 + 4)
\nNow,
\nBy using the Distributive Property,
\n2 (3 + 4) = 2 \u00d7 (3 + 4)
\n= (2 \u00d7 3) + (2 \u00d7 4)
\nHence, from the above,
\nWe can conclude that
\nThe given expression could be described as a product of two factors<\/p>\n

Question 5.
\nWhich part of the expression 2(3 + 4) is the sum of two terms? Explain.
\nAnswer:
\nThe given expression is: 2 (3 + 4)
\nNow,
\nWe know that,
\nEach part of an expression that is separated by a plus sign or a minus sign is called a “Term”
\nSo,
\nIn the given expression,
\nWe can observe that
\n(3 + 4) contains two terms separated by an addition operation
\nHence, from the above,
\nWe can conclude that
\nIn the given expression,
\n(3 + 4) is the sum of two terms<\/p>\n

Do You Know How?<\/strong><\/p>\n

In 6 and 7, write an algebraic expression for each situation.<\/strong><\/p>\n

Question 6.
\nfive less than y
\nAnswer:
\nThe given situation is: five less than y
\nNow,
\nWe know that,
\nWhen we see the word “Less” in a situation, the operation that takes place in that situation must be “Subtraction”
\nHence, from the above,
\nWe can conclude that
\nThe representation of the given situation into an algebraic expression is: y – 5<\/p>\n

Question 7.
\nsix times the quantity two x plus three y
\nAnswer:
\nThe given situation is: six times the quantity two x plus three y
\nNow,
\nWe know that,
\nWhen we see the word “times the quantity” in a situation, the operation that takes place in that situation must be “Multiplication”
\nHence, from the above,
\nWe can conclude that
\nThe representation of the given situation into an algebraic expression is: 6 (2x + 3y)<\/p>\n

In 8-10, use the expression \\(\\frac{w}{4}\\) + 12.5 \u2013 7z.<\/strong><\/p>\n

8. How many terms does the expression have?
\nAnswer:
\nThe given expression is: \\(\\frac{w}{4}\\) + 12.5 – 7z
\nNow,
\nWe know that,
\nEach part of an expression that is separated by a plus sign or a minus sign is called a “Term”
\nHence, from the above,
\nWe can conclude that
\nThe number of terms the given expression contains: 3 terms<\/p>\n

Question 9.
\nWhich term has a coefficient? Explain.
\nAnswer:
\nThe given expression is: \\(\\frac{w}{4}\\) + 12.5 – 7z
\nNow,
\nWe know that,
\nA “Coefficient” is the number that is multiplied by a variable
\nHence, from the above,
\nWe can conclude that
\nThe term that has a coefficient in the given expression is: 7z<\/p>\n

Question 10.
\nWhich term is a constant numerical value?
\nAnswer:
\nThe given expression is: \\(\\frac{w}{4}\\) + 12.5 – 7z
\nNow,
\nWe know that,
\nA “Constant” is nothing but a numerical value
\nHence, from the above,
\nWe can conclude that
\nThe term that is a constant numerical value in the given expression is: 12.5<\/p>\n

Practice & Problem Solving<\/strong><\/p>\n

In 11-14, write an algebraic expression for each situation.<\/strong><\/p>\n

Question 11.
\n12 times a number g
\nAnswer:
\nThe given situation is: 12 times a number g
\nNow,
\nWe know that,
\nWhen we see the word “times a number” in a situation, the operation that takes place in that situation must be “Multiplication”
\nHence, from the above,
\nWe can conclude that
\nThe representation of the given situation into an algebraic expression is: 12 (g) (or) 12g<\/p>\n

Question 12.
\np pennies added to 22 pennies
\nAnswer:
\nThe given situation is: p pennies added to 22 pennis
\nNow,
\nWe know that,
\nWhen we see the word “Add” in a situation, the operation that takes place in that situation must be “Addition”
\nHence, from the above,
\nWe can conclude that
\nThe representation of the given situation into an algebraic expression is: p + 22<\/p>\n

Question 13.
\n22 divided by a number s
\nAnswer:
\nThe given situation is: 22 divided by a number s
\nNow,
\nWe know that,
\nWhen we see the word “Divide” in a situation, the operation that takes place in that situation must be “Division”
\nHence, from the above,
\nWe can conclude that
\nThe representation of the given situation into an algebraic expression is: 22 \u00f7 s (or) \\(\\frac{22}{s}\\)<\/p>\n

Question 14.
\n12\\(\\frac{3}{4}\\) less than the product of 7 and a number x
\nAnswer:
\nThe given situation is: 12\\(\\frac{3}{4}\\) less than the product of 7 and a number x
\nNow,
\nWe know that,
\nWhen we see the word “Less” and “Product” in a situation, the operation that takes place in that situation must be “Subtraction” and “Multiplication”
\nHence, from the above,
\nWe can conclude that
\nThe representation of the given situation into an algebraic expression is: 7x – 12\\(\\frac{3}{4}\\)<\/p>\n

In 15-18, tell how many terms each expression has.<\/strong><\/p>\n

Question 15.
\n5 – g
\nAnswer:
\nThe given expression is: 5 – g
\nNow,
\nWe know that,
\nEach part of an expression that is separated by a plus sign or a minus sign is called a “Term”
\nHence, from the above,
\nWe can conclude that
\nThe number of terms the given expression contains is: 2 terms<\/p>\n

Question 16.
\n3 + \\(\\frac{1}{2}\\)b
\nAnswer:
\nThe given expression is: 3 + \\(\\frac{1}{2}\\)b
\nNow,
\nWe know that,
\nEach part of an expression that is separated by a plus sign or a minus sign is called a “Term”
\nHence, from the above,
\nWe can conclude that
\nThe number of terms the given expression contains is: 2 terms<\/p>\n

Question 17.
\n\\(\\frac{v}{3}\\) + 2 \u2022 5
\nAnswer:
\nThe given expression is: \\(\\frac{v}{3}\\) + 2 \u2022\u00a05
\nNow,
\nWe know that,
\nEach part of an expression that is separated by a plus sign or a minus sign is called a “Term”
\nHence, from the above,
\nWe can conclude that
\nThe number of terms the given expression contains is: 2 terms<\/p>\n

Question 18.
\n16.2 – (3 \u2022 4) + (14 \u00f7 2)
\nAnswer:
\nThe given expression is: 16.2 – (3 \u2022 4) + (14 \u00f7 2)
\nNow,
\nWe know that,
\nEach part of an expression that is separated by a plus sign or a minus sign is called a “Term”
\nHence, from the above,
\nWe can conclude that
\nThe number of terms the given expression contains is: 3 terms<\/p>\n

In 19 and 20, use the expression 5.3t – (20 \u00f7 4) + 11.<\/strong><\/p>\n

Question 19.
\nWhich part of the expression is a quotient? Describe its parts.
\nAnswer:
\nThe given expression is: 5.3t – (20 \u00f7 4) + 11
\nNow,
\nWe know that,
\nThe “Quotient” is only possible when there is a “Division” operation
\nHence, from the above,
\nWe can conclude that
\nThe part of the expression that is a quotient is: 20 \u00f7 4<\/p>\n

Question 20.
\nWhich part of the expression is a product of two factors? Describe its parts.
\nAnswer:
\nThe given expression is: 5.3t – (20 \u00f7 4) + 11
\nNow,
\nWe know that,
\nThe “product” is possible when there are 2 numbers or a number and a variable
\nHence, from the above,
\nWe can conclude that
\nThe part of the expression that is a product of two factors is: 5.3t<\/p>\n

In 21 and 22, use the table at the right.
\n\"Envision
\n<\/strong><\/p>\n

Question 21.
\nModel with Math<\/strong>
\nWrite an expression to show how much longer the round-trip to San Diego is than the round-trip to San Jose. How many terms does the expression have?
\nAnswer:
\nThe given information is:
\n\"\"
\nNow,
\nFrom the given table,
\nWe can observe that
\nThe distance of the round trip to San Jose is: 236 miles
\nThe distance of the round trip to San Diego is: 1,012 miles
\nSo,
\nThe expression that shows how much longer the round-trip to San Diego is than the round-trip to San Jose
\n= (The distance of the round trip to San Diego) – (The distance of the round trip to San Jose)
\n= 1,012 – 236
\nNow,
\nWe know that,
\nEach part of an expression that is separated by a plus sign or a minus sign is called a “Term”
\nSo,
\nThe number of terms in the above expression is: 2 terms
\nHence, from the above,
\nWe can conclude that
\nThe expression that shows how much longer the round-trip to San Diego is than the round-trip to San Jose is: 1,012 – 236
\nThe number of terms in the above expression is: 2 terms<\/p>\n

Question 22.
\nMake Sense and Persevere<\/strong> Last month, a truck driver made 5 round-trips to Los Angeles and some round-trips to San Diego. Write an expression that shows how many miles he drove in all. Identify and describe the part of the expression that shows how many miles he drove and trips he made to San Diego.
\nAnswer:
\nIt is given that
\nLast month, a truck driver made 5 round-trips to Los Angeles and some round-trips to San Diego
\nNow,
\nThe given information is:
\n\"\"
\nNow,
\nFrom the above table,
\nWe can observe that
\nThe distance of the round trip to Los Angeles is: 770 miles
\nThe distance of the round trip to San Diego is: 1,012 miles
\nNow,
\nLet the number of round trips to San Diego be: x
\nLet the number of miles the truck driver drove to Los Angeles be: p
\nLet the number of miles the truck driver drove to San Diego be: q
\nNow,
\nWe know that,
\nThe total distance traveled by the truck driver = (The number of round trips made to Los Angeles) \u00d7 (The number of miles the truck driver drove to Los Angeles) + (The number of round trips to San Diego) \u00d7 (The number of miles the truck driver drove to San Diego)
\n= 5p + xq
\nSo,
\nThe number of miles the truck driver drove is: 5p + xq
\nThe number of trips the truck driver made to san Diego is: x
\nHence, from the above,
\nWe can conclude that
\nThe number of miles the truck driver drove is: 5p + xq
\nThe number of trips the truck driver made to san Diego is: x<\/p>\n

Question 23.
\nUse the expression y \u00f7 3(4 – 2) + 5.5 to complete the table. Identify the parts of the expression that correspond to the descriptions.
\n\"Envision
\nAnswer:
\nThe given expression is: y \u00f7 3 (4 – 2) + 5.5
\nNow,
\nWe know that,
\nA “Variable”, written as a letter, represents a quantity that can change
\nThe symbol of Difference is: –
\nThe symbol of Product is () (or) \u00d7
\nHence, from the above,
\nWe can conclude that
\nThe completed table with the corresponding descriptions is:
\n\"\"<\/p>\n

Question 24.
\nThe floats in the Orlando Citrus parade may use as many citrus fruits as a small orchard produces in 6 years. If f is the number of citrus fruits a small orchard produces in 1 year, write an algebraic expression to represent the number of citrus fruits the floats in the parade may use.
\nAnswer:
\nIt is given that
\nThe floats in the Orlando Citrus parade may use as many citrus fruits as a small orchard produces in 6 years and f is the number of citrus fruits a small orchard produces in 1 year
\nSo,
\nThe number of citrus fruits the floats in the parade may use in 6 years = 6 \u00d7 (The number of citrus fruits a small orchard produces in 1 year)
\n= 6 \u00d7 f
\n= 6f citrus fruits
\nHence, from the above,
\nWe can conclude that
\nThe number of citrus fruits the floats in the parade may use in 6 years is: 6f citrus fruits<\/p>\n

Question 25.
\nCritique Reasoning<\/strong> Anthony says that the expression abc has three terms because it uses three different variables. Critique Anthony’s reasoning and explain whether he is correct
\nAnswer:
\nIt is given that
\nAnthony says that the expression abc has three terms because it uses three different variables.
\nNow,
\nThe given expression is: abc
\nNow,
\nWe know that,
\nEach part of an expression that is separated by a plus sign or a minus sign is called a “Term”
\nNow,
\nFrom the given expression,
\nWe can observe that
\nThere are not any plus or minus signs
\nSo,
\nThe given expression is a single term
\nHence, from the above,
\nWe can conclude that
\nAnthony’s reasoning is incorrect<\/p>\n

Question 26.
\nYuri walked p poodles and b bulldogs on Monday. He walked the same number of poodles and bulldogs each day Tuesday through Friday as he did on Monday. Write an algebraic expression to represent how many total dogs were walked in this 5-day period.
\n\"Envision
\nAnswer:
\nIt is given that
\nYuri walked p poodles and b bulldogs on Monday. He walked the same number of poodles and bulldogs each day Tuesday through Friday as he did on Monday
\nSo,
\nThe total dogs walked on Monday = (The number of poodles) + (The number of bulldogs)
\n= p + b
\nNow,
\nIt is given that
\nThe same number of dogs walked from Tuesday to Friday as on Monday
\nSo,
\nThe total number of dogs that were waked from Monday to Friday = (The number of days from Monday to Friday) \u00d7 (The number of dogs walked on Monday)
\n= 5 \u00d7 (p + b) dogs
\nHence, from the above,
\nWe can conclude that
\nThe total number of dogs that were waked from Monday to Friday is: 5(p + b) dogs<\/p>\n

Question 27.
\nHigher-Order Thinking<\/strong> Some students equally share 2 baskets of oranges. Each basket has 12 oranges. Write an algebraic expression to represent this situation. Then explain how you chose which variable and operations to use.
\nAnswer:
\nIt is given that
\nSome students equally share 2 baskets of oranges. Each basket has 12 oranges
\nNow,
\nLet the number of students be: x
\nLet the number of apples each student gets be: y
\nNow,
\nThe total number of oranges in 2 baskets = 2(The number of baskets in each basket)
\n= 2 (12)
\n= 24 oranges
\nSo,
\nThe algebraic expression that represents the given situation is:
\nThe number of apples each student gets = (The total number of apples in 2 baskets) \u00f7 (The number of students)
\ny = \\(\\frac{24}{x}\\)
\nHence, from the above,
\nWe can conclude that
\nThe algebraic expression that represents the given situation is:
\ny = \\(\\frac{24}{x}\\)<\/p>\n

Question 28.
\nModel with Math<\/strong>
\nThe figure at the right is a regular octagon with side length s. Write two algebraic expressions that use different operations to represent the perimeter of the figure.
\n\"Envision
\nAnswer:
\nIt is given that
\nThe figure at the right is a regular octagon with side length s.
\nNow,
\nThe given figure is:
\n\"Envision
\nNow,
\nWe know that,
\nThe “Perimeter” of any figure is the sum of all the sides of the given figure
\nNow,
\nLet the perimeter of the regular Octagon be: P
\nSo,
\nThe perimeter of the given regular Octagon is:
\nP = s + s + s + s + s + s + s + s
\nP = 8 (s) = 8s
\nHence, from the above,
\nWe can conclude that
\nThe two algebraic expressions that use different operations to represent the perimeter of the given figure is:
\nP = s + s + s + s + s + s + s + s
\nP = 8s<\/p>\n

Assessment Practice<\/strong><\/p>\n

Question 29.
\nWhich algebraic expression represents the phrase Four more than the product 3 times the number of c cats?
\nA. 4 + 3c
\nB. (4 + 3)c
\nC. 3 + 4c
\nD. 4 \u00d7 3 \u00d7 c
\nAnswer:
\nThe given phrase is: Four more than the product 3 times the number of c cats
\nSo,
\nThe representation of the given phrase in the form of an algebraic expression is: 4 + 3c
\nHence, from the above,
\nWe can conclude that
\nThe algebraic expression represents the phrase Four more than the product 3 times the number of c cats is:
\n\"\"<\/p>\n

Question 30.
\nSelect all of the phrases that could be represented by the algebraic expression \\(\\frac{w}{4}\\) – 4.
\n\"Envision four less than the quotient of a number w and four
\n\"Envision the difference between a number w and 4
\n\"Envision four less than w divided by 4
\n\"Envision four less than a number w
\n\"Envision the quotient of four and a number w
\nAnswer:
\nThe given algebraic expression is: \\(\\frac{w}{4}\\) – 4
\nSo,
\nThe representation of the given algebraic expression in the form of a phrase is:
\na. 4 less than the quotient of w and 4
\nb. The difference between the quotient of w and 4 and 4
\nc. 4 less than w divided by 4
\nHence, from the above,
\nWe can conclude that
\nAll of the phrases that could be represented by the algebraic expression \\(\\frac{w}{4}\\) – 4. are:
\n\"\"<\/p>\n

Lesson 3.5 Evaluate Algebraic expressions<\/h3>\n

Solve & Discuss It!<\/strong><\/p>\n

A bike shop charges by the hour to rent a bike. Related items are rented for flat fees. Write an expression that represents how much it will cost to rent a bike and helmet for h hours. How much would it cost to rent a bike and a helmet for 3 hours?
\nModel with Math<\/strong>
\nYou can write an algebraic expression with decimals in the same way you do with whole numbers.
\n\"Envision<\/p>\n

Answer:
\nIt is given that
\nA bike shop charges by the hour to rent a bike. Related items are rented for flat fees
\nNow,
\nThe given table is:
\n\"\"
\nSo,
\nThe total cost of renting a bike and a helmet for h hours = (The cost of renting a bike per hour) \u00d7 (The total number of hours) + The cost of a helmet
\n= $12.50 \u00d7h + $5.25
\nSo,
\nThe total cost of renting a bike and a helmet for 3 hours = (The cost of renting a bike per hour) \u00d7 (The total number of hours) + The cost of a helmet
\n= $12.50 \u00d73 + $5.25
\n= $37.50 + $5.25
\n= $42.75
\nHence, from the above,
\nWe can conclude that
\nThe expression that represents the total cost to rent a bike and a helmet for h hours is:
\nThe total cost of renting a bike and a helmet for h hours =$12.50 \u00d7h + $5.25
\nThe total cost of renting a bike and a helmet for 3 hours is: $42.75<\/p>\n

Focus on math practices<\/strong>
\nUse Structure<\/strong> Write an expression that represents renting a bike, a lock, and a basket for h hours. What is the cost of renting this equipment for 4 hours?
\nAnswer:
\nThe given table is:
\n\"\"
\nNow,
\nFrom the given table,
\nWe can observe that
\nThe cost to rent a bike, a lock, and a basket for h hours = (The cost of renting a bike per hour) \u00d7 (The total number of hours) + The cost of renting a lock + The cost of renting a basket
\n= $12.50 \u00d7 h + $1.75 + $2.25
\n= $12.50 \u00d7 h + 4
\nNow,
\nThe cost to rent a bike, a lock, and a basket for 4 hours = (The cost of renting a bike per hour) \u00d7 (The total number of hours) + The cost of renting a lock + The cost of renting a basket
\n= $12.50 \u00d7 4 + $1.75 + $2.25
\n= $12.50 \u00d7 4 + $4
\n= $50 + $4
\n= $54
\nHence, from the above,
\nWe can conclude that
\nThe expression that represents renting a bike, a lock, and a basket for h hours is:
\nThe expression that represents renting a bike, a lock, and a basket for h hours = $12.50 \u00d7 h + 4
\nThe cost to rent a bike, a lock, and a basket for 4 hours is: $54<\/p>\n

VISUAL LEARNING<\/strong><\/p>\n

? Essential Question<\/strong> How can you evaluate an algebraic expression?
\nAnswer:
\nTo evaluate an algebraic expression means to determine the value of the expression for a given value of each variable in\u00a0the expression. Replace each variable in the expression with the given value, then simplify the resulting expression using the order of operations.<\/p>\n

Try It!<\/strong>
\nEvaluate the expression 50 – t when t equals 10, 20, or 25. Then complete the table to show the values.
\n\"Envision
\nAnswer:
\nThe given expression is: 50 – t
\nWhere,
\nThe value of t is: 10, 20, and 25
\nNow,
\nFor t = 10, 20, and 25,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n\"\"<\/p>\n

Convince Me!<\/strong> What does it mean to use substitution to evaluate an algebraic expression?
\nAnswer:
\nTo substitute the values of the\u00a0numerals into an algebraic expression means to replace all numerals with their respective values (or numbers). Once the values of the numerals are substituted, the expression can be evaluated. While evaluating, the correct order of operations must be observed<\/p>\n

Try It!<\/strong>
\nEvaluate the expression 3.4 + 12a \u00f7 4 for a = 10.
\nAnswer:
\nThe given expression is 3.4 + 12a \u00f7 4 for a = 10
\nNow,
\nFor a = 10,
\n3.4 + 12a \u00f74
\n= 3.4 + 12 (10) \u00f7 4
\n= 3.4 + 120 \u00f7 4
\n= 3.4 + 30
\n= 33.4
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: 33.4<\/p>\n

Try It!<\/strong>
\nSuppose Mr. Grant decides to buy square tiles that have side lengths of \\(\\frac{3}{4}\\) foot. How many of these tiles will he need to buy?
\nAnswer:
\nIt is given that
\nSuppose Mr. Grant decides to buy square tiles that have side lengths of \\(\\frac{3}{4}\\) foot
\nNow,
\nFrom Example 3,
\nIt is given that
\nThe expression to find the number of tiles Mr. Grant needs to buy is 27 \u00f7 s\u00b2
\nWhere,
\ns is the sidelength
\nNow,
\nWhen s = \\(\\frac{3}{4}\\) foot,
\nThe number of tiles Mr.Grant needs to buy = 27 \u00f7 (\\(\\frac{3}{4}\\))\u00b2
\n= 27 \u00f7 \\(\\frac{9}{16}\\)
\n= 27 \u00d7 \\(\\frac{16}{9}\\)
\n= \\(\\frac{27 \u00d7 16}{9}\\)
\n= 3 \u00d7 16
\n= 48 tiles
\nHence, from the above,
\nWe can conclude that
\nThe number of tiles Mr.Grant needed to buy when the side length is \\(\\frac{3}{4}\\) foot is: 48 tiles<\/p>\n

KEY CONCEPT<\/strong>
\nTo evaluate an expression, use substitution to replace a variable with its numerical value. Then use the order of operations to simplify.
\n\"Envision<\/p>\n

Do You Understand?<\/strong><\/p>\n

Question 1.
\n? Essential Question<\/strong> How can you evaluate an algebraic expression?
\nAnswer:
\nTo evaluate an algebraic expression means to determine the value of the expression for a given value of each variable in\u00a0the expression. Replace each variable in the expression with the given value, then simplify the resulting expression using the order of operations<\/p>\n

Question 2.
\nConstruct Arguments<\/strong> Why is it important to use the order of operations to evaluate algebraic expressions?
\nAnswer:
\nThe order of operations is a rule that tells you the right order in which to solve different parts of a math problem. The order of operations is important because it guarantees that people can all read and solve a problem in the same way<\/p>\n

Question 3.
\nHow is evaluating an expression with fractions like evaluating an expression with whole numbers? How is it different?
\nAnswer:
\nThe fractions have the same denominators so you can start by adding the first two fractions. Add the fractions before adding the whole numbers. Then, take the sum and subtract the third fraction. Before evaluating the expression, rewrite the fractions with one common denominator
\nTo evaluate an expression with a whole number, we substitute the given number for the variable in the expression and then simplify the expression using the order of operations. in the expression that involves fractions, first, we have to simplify the fractions into whole numbers and follow the order of operations to evaluate them<\/p>\n

Question 4.
\nReasoning<\/strong> Annalise earns $4 an hour walking pets in her neighborhood. She evaluates the expression 4h, where h represents the number of hours, to find the amount she earns. Can any number be substituted for h? Explain.
\nAnswer:
\nIt is given that
\nAnnalise earns $4 an hour walking pets in her neighborhood. She evaluates the expression 4h, where h represents the number of hours, to find the amount she earns.
\nNow,
\nAccording to the given information,
\nThe total amount of money earned by Annalise = $4h
\nWhere,
\nh is the number of hours
\nNow,
\nWe know that,
\nh will be any number i.e., either a whole number or a fraction
\nHence, from the above,
\nWe can conclude that
\nWe can substitute any number for h<\/p>\n

Do You Know How?<\/strong><\/p>\n

In 5-8, evaluate each expression for t = 8, w = \\(\\frac{1}{2}\\), and x = 3.<\/strong><\/p>\n

Question 5.
\n3t – 8
\nAnswer:
\nThe givene xpression is: 3t – 8
\nNow,
\nFor t = 8,
\n3t – 8
\n= 3 (8) – 8
\n= 24 – 8
\n= 16
\nHence, from the baove,
\nWe can conclude that
\nThe value of the given expression is: 16<\/p>\n

Question 6.
\n6w \u00f7 x + 9
\nAnswer:
\nThe given expression is: 6w \u00f7 x + 9
\nNow,
\nFor w = \\(\\frac{1}{2}\\), and x = 3,
\n6w \u00f7 x + 9
\n= 6 (\\(\\frac{1}{2}\\)) \u00f7 3 + 9
\n= \\(\\frac{6}{2}\\) \u00f7 3 + 9
\n= 3 \u00f7 3 + 9
\n= 1 + 9
\n= 10
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: 10<\/p>\n

Question 7.
\nt2<\/sup> \u2013 12w \u00f7 \u00d7
\nAnswer:
\nThe given expression is: t2<\/sup> \u2013 12w \u00f7 \u00d7
\nNow,
\nFor t = 8, w = \\(\\frac{1}{2}\\), and x = 3,
\nt2<\/sup> \u2013 12w \u00f7 \u00d7
\n= 82<\/sup> \u2013 12 (\\(\\frac{1}{2}\\)) \u00f7 3
\n= 64 – \\(\\frac{12}{2}\\) \u00f7 3
\n= 64 – 6 \u00f7 3
\n= 64 – 2
\n= 62
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: 62<\/p>\n

Question 8.
\n5x \u2013 2w + t
\nAnswer:
\nThe given expression is: 5x – 2w + t
\nNow,
\nFor t = 8, w = \\(\\frac{1}{2}\\), and x = 3,
\n5x – 2w + t
\n= 5 (3) – 2 (\\(\\frac{1}{2}\\)) + 8
\n= 15 – \\(\\frac{2}{2}\\) + 8
\n= 15 – 1 + 8
\n= 15 + 7
\n= 22
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: 22<\/p>\n

In 9-14, evaluate each expression for the value given.<\/strong><\/p>\n

Question 9.
\nz \u00f7 4; z = 824
\nAnswer:
\nThe given expression is: z \u00f7 4 with z = 824
\nNow,
\nFor z = 824,
\nz \u00f7 4
\n= 824 \u00f7 4
\n= 206
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: 206<\/p>\n

Question 10.
\n6t \u00f7 9 – 22; t = 60
\nAnswer:
\nThe given expression is: 6t \u00f7 9 – 22 with t = 60
\nNow,
\nFor t = 60,
\n6t \u00f7 9 – 22
\n= 6 (60) \u00f7 9 – 22
\n= 360 \u00f7 9 – 22
\n= 40 – 22
\n= 18
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: 18<\/p>\n

Question 11.
\nr \u00f7 2.4; r = 16.8
\nAnswer:
\nThe given expression is: r \u00f72.4 with r = 16.8
\nNow,
\nFor r = 16.8,
\nr \u00f72.4
\n= 16.8 \u00f7 2.4
\n= 7
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: 7<\/p>\n

Question 12.
\n9.85 \u00d7 s; s = 4
\nAnswer:
\nThe given expression is: 9.85 \u00d7 s with s = 4
\nNow,
\nFor s = 4,
\n9.85 \u00d7 s
\n= 9.85 \u00d7 4
\n= 39.40
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: 39.40<\/p>\n

Question 13.
\nx \u00f7 12; x = \\(\\frac{2}{3}\\)
\nAnswer:
\nThe given expression is: x \u00f7 12, with x = \\(\\frac{2}{3}\\)
\nNow,
\nFor x = \\(\\frac{2}{3}\\),
\nx \u00f7 12
\n= \\(\\frac{2}{3}\\) \u00f7 12
\n= \\(\\frac{2}{3}\\) \u00d7 \\(\\frac{1}{12}\\)
\n= \\(\\frac{2 \u00d71}{3 \u00d7 12}\\)
\n= \\(\\frac{2}{36}\\)
\n= \\(\\frac{1}{18}\\)
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: \\(\\frac{1}{18}\\)<\/p>\n

Question 14.
\n\\(\\frac{3}{4}\\) + 4y \u00f7 3; y = 1\\(\\frac{1}{2}\\)
\nAnswer:
\nThe given expression is: \\(\\frac{3}{4}\\) + 4y \u00f7 3 with y = 1\\(\\frac{1}{2}\\)
\nNow,
\nFor y = 1\\(\\frac{1}{2}\\),
\n\\(\\frac{3}{4}\\) + 4y \u00f7 3
\n= \\(\\frac{3}{4}\\) + 4 (1\\(\\frac{1}{2}\\) \u00f7 3
\n= \\(\\frac{3}{4}\\) + 3 \u00f7 3
\n= \\(\\frac{3}{4}\\) + 1
\n= \\(\\frac{7}{4}\\)
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: \\(\\frac{7}{4}\\)<\/p>\n

Practice & Problem Solving<\/strong><\/p>\n

Scan for Multimedia<\/strong><\/p>\n

In 15-17, evaluate each expression for w = 5, x = 3, y = 4, and z = 8.<\/strong><\/p>\n

Question 15.
\n9x
\nAnswer:
\nThe given expression is: 9x
\nNow,
\nFor x = 3,
\n9x = 9 \u00d7 3
\n= 27
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: 27<\/p>\n

Question 16.
\n3y + 6 \u00f7 2x
\nAnswer:
\nThe given expression is: 3y + 6 \u00f7 2x
\nNow,
\nFor x = 3, and y = 4,
\n3y + 6 \u00f7 2x
\n= (3 \u00d7 4) + 6 \u00f7 (2 \u00d7 3)
\n= 12 + 6 \u00f7 6
\n= 12 + 1
\n= 13
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: 13<\/p>\n

Question 17.
\nw2<\/sup> + 2 + 48 \u00f7 2z
\nAnswer:
\nThe given expression is: w2<\/sup> + 2 + 48 \u00f7 2z
\nNow,
\nFor w = 5, and z = 8,
\nw2<\/sup> + 2 + 48 \u00f7 2z
\n= 5\u00b2 + 2 + 48 \u00f7 (2 \u00d7 8)
\n= 25 + 2 + 48 \u00f7 16
\n= 27 + 3
\n= 30
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: 30<\/p>\n

In 18-20, evaluate each expression for x = 1.8, x = 5, and x = 6.4.<\/strong><\/p>\n

Question 18.
\nx \u00f7 4
\nAnswer:
\nThe given expression is: x \u00f7 4
\nNow,
\nFor x = 1.8
\nx \u00f7 4 = 1.8 \u00f7 4
\n= 0.45
\nNow,
\nFor x = 5,
\nx \u00f7 4 = 5 \u00f7 4
\n= 1.25
\nNow,
\nFor x = 6.4,
\nx \u00f7 4 = 6.4 \u00f7 4
\n= 1.6
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression for x = 1.8 is: 0.45
\nThe value of the given expression for x = 5 is: 1.25
\nThe value of the given expression for x = 6.4 is: 1.6<\/p>\n

Question 19.
\nx(3.35)
\nAnswer:
\nThe given expression is: x (3.35)
\nNow,
\nFor x = 1.8,
\nx (3.35) = 1.8 (3.35)
\n= 6.03
\nNow,
\nFor x = 5,
\nx (3.35) = 5 (3.35)
\n= 16.75
\nNow,
\nFor x = 6.4,
\nx (3.35) = 6.4 (3.35)
\n= 21.44
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression for x = 1.8 is: 6.03
\nThe value of the given expression for x = 5 is: 16.75
\nThe value of the given expression for x = 6.4 is: 21.44<\/p>\n

Question 20.
\n2x + 3.1
\nAnswer:
\nThe given expression is: 2x + 3.1
\nNow,
\nFor x = 1.8,
\n2x + 3.1 = (2 \u00d7 1.8) + 3.1
\n= 6.7
\nNow,
\nFor x = 5,
\n2x + 3.1 = (2 \u00d7 5) + 3.1
\n= 10 + 3.1
\n= 13.1
\nNow,
\nFor x = 6.4,
\n2x + 3.1 = (2 \u00d7 6.4) + 3.1
\n= 15.9
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression for x = 1.8 is: 6.7
\nThe value of the given expression for x = 5 is: 13.1
\nThe value of the given expression for x = 6.4 is: 15.9<\/p>\n

In 21-23, evaluate each expression for the value given.<\/strong><\/p>\n

Question 21.
\nj + \\(\\frac{3}{8}\\); j = \\(\\frac{3}{4}\\)
\nAnswer:
\nThe given expression is: j + \\(\\frac{3}{8}\\)
\nNow,
\nFor j = \\(\\frac{3}{4}\\),
\nj + \\(\\frac{3{8}\\)
\n= \\(\\frac{3}{8}\\) + \\(\\frac{3}{4}\\)
\n= \\(\\frac{3}{8}\\) + \\(\\frac{6}{8}\\)
\n= \\(\\frac{3 + 6}{8}\\)
\n= \\(\\frac{9}{8}\\)
\nHence, from the baove,
\nWe can conclude that
\nThe value of the given expression is: \\(\\frac{9}{8}\\)<\/p>\n

Question 22.
\n8 – g \u00f7 \\(\\frac{7}{8}\\); g = \\(\\frac{5}{6}\\)
\nAnswer:
\nThe given expression is: 8 – g \u00f7 \\(\\frac{7}{8}\\)
\nNow,
\nFor g = \\(\\frac{5}{6}\\),
\n8 – g \u00f7 \\(\\frac{7}{8}\\)
\n= 8 – \\(\\frac{5}{6}\\) \u00f7 \\(\\frac{7}{8}\\)
\n= 8 – \\(\\frac{5}{6}\\) \u00d7 \\(\\frac{8}{7}\\)
\n= 8 – \\(\\frac{5 \u00d7 8}{6 \u00d7 7}\\)
\n= 8 – \\(\\frac{40}{42}\\)
\n= 8 – \\(\\frac{20}{21}\\)
\n= \\(\\frac{168 – 20}{21}\\)
\n= \\(\\frac{148}{21}\\)
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: \\(\\frac{148}{21}\\)<\/p>\n

Question 23.
\n3m \u00f7 \\(\\frac{2}{5}\\); m = \\(\\frac{2}{3}\\)
\nAnswer:
\nThe given expression is: 3m \u00f7 \\(\\frac{2}{5}\\)
\nNow,
\nFor m = \\(\\frac{2}{3}\\),
\n3 \u00d7 \\(\\frac{2}{3}\\) \u00f7 \\(\\frac{2}{5}\\)
\n= 2 \u00f7 \\(\\frac{2}{5}\\)
\n= 2 \u00d7 \\(\\frac{5}{2}\\)
\n= 5
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: 5<\/p>\n

Question 24.
\nEvaluate the expression for the values of b.
\n\"Envision
\nAnswer:
\nThe given expression is: b (3) + 20.4
\nNow,
\nFor b = 8.9,
\nb (3) + 20.4
\n= 8.9 (3) + 20.4
\n= 47.1
\nNow,
\nFor b = 5.1,
\nb (3) + 20.4
\n= 5.1 (3) + 20.4
\n= 35.7
\nNow,
\nFor b = 0.2,
\nb (3) + 20.4
\n= 0.2 (3) + 20.4
\n= 21
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression for the given values of b is:
\n\"\"<\/p>\n

Question 25.
\nEvaluate the expression for the values of j.
\n\"Envision
\nAnswer:
\nThe given expression is: 2j + \\(\\frac{3}{5}\\)
\nNow,
\nFor j = \\(\\frac{1}{2}\\),
\n2j + \\(\\frac{3}{5}\\)
\n= 2 (\\(\\frac{1}{2}\\)) + \\(\\frac{3}{5}\\)
\n= 1 + \\(\\frac{3}{5}\\)
\n= \\(\\frac{8}{5}\\)
\n= 1.6
\nNow,
\nFor j = \\(\\frac{4}{5}\\),
\n2j + \\(\\frac{3}{5}\\)
\n= 2 (\\(\\frac{4}{5}\\)) + \\(\\frac{3}{5}\\)
\n= \\(\\frac{8}{5}\\) + \\(\\frac{3}{5}\\)
\n= \\(\\frac{11}{5}\\)
\n= 2.2
\nFor j = 1\\(\\frac{3}{4}\\),
\n2j + \\(\\frac{3}{5}\\)
\n= 2 (1\\(\\frac{3}{4}\\)) + \\(\\frac{3}{5}\\)
\n= \\(\\frac{14}{4}\\) + \\(\\frac{3}{5}\\)
\n= \\(\\frac{82}{20}\\)
\n= 4.1
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression for the values of j is:
\n\"\"<\/p>\n

In 26-28, use the table at the right.<\/strong><\/p>\n

Question 26.
\nModel with Math<\/strong>
\na. Ms. White wants to rent a small car for a week. It will cost the weekly fee plus $0.30 per mile driven. Let m= the number of miles Ms. White drives during the week. Write an expression that shows the amount she will pay for the car.
\n\"Envision
\nAnswer:
\nIt is given that
\nMs. White wants to rent a small car for a week. It will cost the weekly fee plus $0.30 per mile driven. Let m= the number of miles Ms. White drives during the week.
\nNow,
\nThe given table is:
\n\"\"
\nSo,
\nThe total amount Ms. White will pay for the car = (The rent of a small car for a week) + (The extra fee for a small car per mile during the week) \u00d7 (The total number of miles)
\n= $250 + $0.30m
\nHence, from the above,
\nWe can conclude that
\nThe expression that represents the total amount Ms. White will pay for the car is: $250 + $0.30m<\/p>\n

b. Evaluate the expression you wrote to find how much Ms. White will pay if she drives 100 miles.
\nAnswer:
\nFrom Part (a),
\nWe can observe that
\nThe expression that represents the total amount Ms. White will pay for the car is: $250 + $0.30m
\nNow,
\nWhen Ms. White drives for 100 miles,
\nThe total amount Ms. White will pay for the car if she drives 100 miles = $250 + $0.30 (100)
\n= $250 + $30
\n= $280
\nHence, from the above,
\nWe can conclude that
\nThe total amount Ms. White will pay for the car if she drives 100 miles is: $280<\/p>\n

Question 27.
\nMr. Black rents a lu\u00d7ury car for one week and a few days, d. He does not pay a per-mile fee. Evaluate the expression 325 + 120d to find how much Mr. Black will pay for an 11-day rental.
\nAnswer:
\nIt is given that
\nMr. Black rents a lu\u00d7ury car for one week and a few days, d. He does not pay a per-mile fee.
\nNow,
\nThe given table is:
\n\"\"
\nNow,
\nAccording to the given information,
\nThe expression that represents the total amount paid by Mr.Black for one week and a few days is: $325 + $120d
\nWhere
\nd is the number of few days
\nNow,
\nFor an 11-day rental,
\nThe total amount paid by Mr.Black for one week and a few days = $325 + $120d
\n= $325 + $120 (11)
\n– $325 + $132
\n= $457
\nHence, from the above,
\nWe can conclude that
\nThe total amount Mr.Black will pay for an 11-day rental is: $457<\/p>\n

Question 28.
\nFor any of the vehicles listed in the table, how many days can you rent the vehicle before it would be less expensive to rent for the week?
\nAnswer:
\nThe given list of vehicles is:
\n\"\"
\nNow,
\nWhen we observe the list,
\nThe least expensive rent is: $250 for a week
\nNow,
\nWhen we observe the rents for a day,
\nThere are only 2 days’ rents that are less than the least expensive rent for a week
\nHence, from the above,
\nWe can conclude that
\nWe can rent the vehicle for 2 days before it would be less expensive to rent for the week<\/p>\n

In 29 and 30, use the table at the right.<\/strong><\/p>\n

Question 29.
\nModel with Math<\/strong>
\nTamara is making a medium-length necklace. Write an expression that shows how much it will cost Tamara for the chain, pendant, and b beads that cost $0.25 each. Then find the total cost of the necklace if Tamara uses 30 beads.
\n\"Envision
\nAnswer:
\nIt is given that
\nTamara is making a medium-length necklace
\nNow,
\nThe given table is:
\n\"\"
\nSo,
\nThe total cost of the medium-length necklace made by Tamara = (The cost of medium length necklace) + (The cost of medium length pendant) + (The cost of each bead) \u00d7 (The total number of beads)
\n= $1.80 + $3.72 + $0.25b
\n= $5.52 + $0.25b
\nNow,
\nWhen there are 30 beads,
\nThe total cost of the medium length necklace made by Tamara when there are 30 beads = $5.52 + $0.25b
\n= $5.52 + $0.25 (30)
\n= $13.02
\nHence, from the above,
\nWe can conclude that
\nThe total cost of the medium-length necklace made by Tamara, when there are 30 beads, is: $13.02<\/p>\n

Question 30.
\nHigher-Order Thinking<\/strong> Ronnie is making short and long necklaces with only one chain and one pendant per necklace. Write an expression that shows how much it will cost Ronnie to make s short necklaces and n long necklaces. Then find the cost for 3 short necklaces and 2 long necklaces.
\nAnswer:
\nIt is given that
\nRonnie is making short and long necklaces with only one chain and one pendant per necklace.
\nNow,
\nThe given table is:
\n\"\"
\nNow,
\nThe cost of a necklace made by Ronnie = [(The cost of a short-chain) + (The cost of a short pendant)] \u00d7 (The number of short necklaces) + [(The cost of a long chain) + (The cost of a long pendant)] \u00d7 (The number of long necklaces)
\n= ($1.15 + $2.39) \u00d7 s + ($2.25 + $4.50) \u00d7 n
\n= $3.54s +\u00a0 $6.75n
\nWhere,
\ns is the number of short necklaces
\nn is the number of long necklaces
\nNow,
\nThe cost of 3 short necklaces = $3.54s
\n= $3.54 \u00d7 3
\n= $10.62
\nNow,
\nThe cost of 2 long necklaces = $6.75n
\n= 6.75 \u00d7 2
\n= $13.50
\nHence, from the above,
\nWe can conclude that
\nThe expression that represents the cost of a necklace made by Ronnie is: $3.54s + $6.75n
\nThe cost of 3 short necklaces is: $10.62
\nThe cost of 2 long necklaces is: $13.50<\/p>\n

Question 31.
\nCritique Reasoning<\/strong> Katrina says that the expression 5,432 + 4,564 + 13,908 = 61n can be evaluated by adding 5,432 + 4,564 + 13,908 and then dividing by the value of 61n. Do you agree? Explain.
\nAnswer:
\nIt is given that
\nKatrina says that the expression 5,432 + 4,564 + 13,908 = 61n can be evaluated by adding 5,432 + 4,564 + 13,908 and then dividing by the value of 61n.
\nNow,
\nThe given expression is: 5,432 + 4,564 + 13,908 = 61n
\nNow,
\nWe have to find the value of n in the given expression,
\nNow,
\nTo find the value of n, the following steps have to be followed:
\nStep 1:
\nAdd 5,432, 4,54, and 13,908
\nStep 2:
\nDivide the result that we obtained in step 1 by 61 to get the value of n
\nBut,
\nAccording to the given information,
\nTo find the value of the given expression, we have to divide the result that we obtained in step 1 by 61n
\nHence, from the above,
\nWe can conclude that
\nWe can not agree with Katrina<\/p>\n

Question 32.
\nThe density, d, of an object can be found by using the formula d = \\(\\frac{m}{V}\\), where m is the mass of the object and visits volume. What is the density of an object that has a mass of 73,430 kilograms and a volume of 7 m3<\/sup>?
\nAnswer:
\nIt is given that
\nThe density, d, of an object can be found by using the formula d = \\(\\frac{m}{v}\\), where m is the mass of the object and visits volume
\nNow,
\nThe given expression is: d = \\(\\frac{m}{V}\\)
\nNow,
\nFor m = 73,430 Kilograms, and V = 7 m\u00b3,
\nd = \\(\\frac{73,430}{7}\\)
\nd = 10,490 Kg \/ m\u00b3
\nHence, from the above,
\nWe can conclude that
\nThe density of the given object for the given values of m and V is: 10,490 Kg \/ m\u00b3<\/p>\n

Question 33.
\nThe formula V = s3<\/sup> can be used to find the volume of a cube. Use the formula to find the volume, V, of a cube-shaped bin with side length s of \\(\\frac{2}{3}\\) yard.
\nAnswer:
\nIt is given that
\nThe formula V = s3<\/sup> can be used to find the volume of a cube.
\nNow,
\nThe given side length of the cube is: \\(\\frac{2}{3}\\) yard
\nSo,
\nV = (\\(\\frac{2}{3}\\))\u00b3
\n= \\(\\frac{2}{3}\\) \u00d7 \\(\\frac{2}{3}\\) \u00d7 \\(\\frac{2}{3}\\)
\n= \\(\\frac{2 \u00d7 2 \u00d7 2}{3 \u00d7 3 \u00d7 3}\\)
\n= \\(\\frac{8}{27}\\) yard\u00b3
\nHence, from the above,
\nWe can conclude that
\nThe volume of the cube-shaped bin with the given side length is: \\(\\frac{8}{27}\\) yard\u00b3<\/p>\n

Question 34.
\nKatie is evaluating the expression 15.75 \u00f7 p + 3p when p = 3.15. Explain each step that she should follow.
\nAnswer:
\nIt is given that
\nKatie is evaluating the expression 15.75 \u00f7 p + 3p when p = 3.15.
\nNow,
\nThe given expression is: 15.75 \u00f7 p + 3p with p = 3.15
\nNow,
\nFor p = 3.15,
\n15.75 \u00f7 p + 3p
\n= 15.75 \u00f7 3.15 + 3 (3.15)
\n= 5 + 9.45
\n= 14.45
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: 14.45<\/p>\n

Assessment Practice<\/strong><\/p>\n

Question 35.
\nAn equation is shown.
\n5x + (x \u00f7 3) = 38.4
\nWhich value of x makes the equation true?
\nA. x = 5.1
\nB. x = 5.2
\nC. x = 6.1
\nD. x = 7.2
\nAnswer:
\nThe given expression is:
\n5x + (x \u00f7 3) = 38.4
\nNow,
\n5x + \\(\\frac{x}{3}\\) = 38.4
\n\\(\\frac{16x}{3}\\) = 38.4
\n16x = 38.4 (3)
\n16x = 115.2
\nx = \\(\\frac{115.2}{16}\\)
\nx = 7.2
\nHence, from the above,
\nWe can conclude that
\nThe value of x that makes the given equation true is:
\n\"\"<\/p>\n

3-Act Mathematical Modeling: The Field Trip<\/strong><\/p>\n

3-ACT MATH<\/strong><\/p>\n

\"Envision<\/p>\n

ACT 1<\/strong><\/p>\n

Question 1.
\nAfter watching the video, what is the first question that comes to mind?
\nAnswer:<\/p>\n

Question 2.
\nWrite the Main Question you will answer.
\nAnswer:<\/p>\n

Question 3.
\nConstruct Arguments<\/strong> Make a prediction to answer this Main Question. Explain your prediction.
\n\"Envision
\nAnswer:<\/p>\n

Question 4.
\nOn the number line below, write a number that is too small to be the answer. Write a number that is too large.
\n\"Envision
\nAnswer:<\/p>\n

Question 5.
\nPlot your prediction on the same number line.
\nAnswer:<\/p>\n

ACT 2<\/strong><\/p>\n

Question 6.
\nWhat information in this situation would be helpful to know? How would you use that information?
\n\"Envision
\nAnswer:<\/p>\n

Question 7.
\nUse Appropriate Tools<\/strong> What tools can you use to solve the problem? Explain how you would use them strategically.
\nAnswer:<\/p>\n

Question 8.
\nModel with Math<\/strong> Represent the situation using mathematics. Use your representation to answer the Main Question.
\nAnswer:<\/p>\n

Question 9.
\nWhat is your answer to the Main Question? Is it higher or lower than your prediction? Explain why.
\n\"Envision
\nAnswer:<\/p>\n

ACT 3<\/strong><\/p>\n

Question 10.
\nWrite the answer you saw in the video.
\n\"Envision
\nAnswer:<\/p>\n

Question 11.
\nReasoning<\/strong> Does your answer match the answer in the video? If not, what are some reasons that would explain the difference?
\nAnswer:<\/p>\n

Question 12.
\nMake Sense and Persevere<\/strong> Would you change your model now that you know the answer? Explain.
\n\"Envision
\nAnswer:<\/p>\n

Reflect<\/strong><\/p>\n

Question 13.
\nModel with Math<\/strong> explain how you used a mathematical model to represent the situation. How did the model help you answer the Main Question?
\nAnswer:<\/p>\n

Question 14.
\nCritique Reasoning<\/strong> A classmate said your model works for any number of students and adults. Do you agree? Justify your reasoning or explain your classmate’s error.
\nAnswer:<\/p>\n

SEQUEL<\/strong><\/p>\n

Question 15.
\nGeneralize<\/strong> Suppose the entire grade goes on the field trip: 283 students and 10 teachers. Each bus holds 72 people and costs $610 for transport. How much money is needed? Explain how you reused your model.
\n\"Envision
\nAnswer:<\/p>\n

Lesson 3.6 Generate Equivalent expressions<\/h3>\n

ACTIVITY<\/strong><\/p>\n

Explain It!<\/strong>
\nJuwon says all three expressions are equivalent.
\n\"Envision<\/p>\n

A. Find the value of each expression for n = 1.
\n\"Envision
\nAnswer:
\nIt is given that
\nJuwon says all three expressions are equivalent.
\nNow,
\nFor n = 1,
\nThe value of each expression will be:
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n\"\"<\/p>\n

B. Find the value of each expression for n = 2.
\n\"Envision
\nAnswer:
\nIt is given that
\nJuwon says all three expressions are equivalent.
\nNow,
\nFor n = 2,
\nThe value of each expression will be:
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n\"\"<\/p>\n

C. Critique Reasoning<\/strong> Do you agree with Juwon that all three expressions are equivalent? Explain.
\nAnswer:
\nAccording to Juwon,
\nThe expressions are:
\na. 8n + 6
\nb. 2 (4n + 3)
\nc. 14n
\nNow,
\nFrom (a),
\n8n + 6 = 2 (4n + 3) \u2260 14n
\nHence, from the above,
\nWe can conclude that
\nWe can not agree with Juwon that all the three expressions are equivalent<\/p>\n

Focus on math practices<\/strong>
\nGeneralize<\/strong> When a number is substituted for the same variable in two expressions, how many times must those two expressions have different values before you know they are not equivalent? Explain.
\nAnswer:
\nWhen a number is substituted for the same variable in two expressions, those two expressions must have\u00a0 2 different values before you know they are not equivalent<\/p>\n

VISUAL LEARNING<\/strong><\/p>\n

? Essential Question<\/strong> How can you identify and write equivalent expressions?
\nAnswer:
\nCombine any like terms on each side of the equation x-terms with x-terms and constants with constants. Arrange the terms in the same order, usually x-term before constants. If all of the terms in the two expressions are identical, then the two expressions are equivalent.<\/p>\n

Try It!<\/strong>
\nWrite an expression that is equivalent to 3y – 9.
\n\"Envision
\nAnswer:
\nThe given expression is: 3y – 9
\nNow,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n3y – 9 is equivalent to 3 (y – 3)<\/p>\n

Convince Me!<\/strong> Why can you use properties of operations to write equivalent expressions?
\nAnswer:
\nThe properties of operations used to write equivalent expressions are:
\na. Commutative Property of Addition: When adding, changing the order of the numbers does not change the sum.
\nb. Commutative Property of Multiplication
\nc. Associative Property of Addition
\nd. Associative Property of Multiplication
\ne. Distributive Property<\/p>\n

Try It!<\/strong>
\nWhich of the following expressions are equivalent? Explain.
\n\"Envision
\nAnswer:
\nThe given expressions are:
\na. 10y + 5\u00a0 \u00a0 b. 15y\u00a0 \u00a0 \u00a0c. 5 (2y + 1)
\nNow,
\n10y + 5 = 5 (2y + 1)
\nHence, from the above,
\nWe can conclude that
\n10y + 5 and 5 (2y + 1) are equivalent expressions<\/p>\n

Try It!<\/strong>
\nAre 2(x \u2013 3) + 1 and 2x + 6 equivalent expressions? Use substitution to justify your work.
\nAnswer:
\nThe given expressions are: 2 (x – 3) + 1 and 2x + 6
\nNow,
\n2 (x – 3) + 1 = 2 (x) – 2 (3) + 1
\n= 2x – 6 + 1
\n= 2x – 5
\nSo,
\n2x – 5 \u2260 2x + 6
\nHence, from the above,
\nWe can conclude that
\n2 (x – 3) + 1 and 2x + 6 are not equivalent expressions<\/p>\n

KEY CONCEPT<\/strong>
\nTwo algebraic expressions are equivalent if they have the same value when any number is substituted for the variable. You can use the properties of operations to write equivalent expressions.<\/p>\n

Properties of Operations<\/strong>
\n1 Commutative Property
\nof Addition a + b = b + a
\nof Multiplication a \u00d7 b = b \u00d7 a
\n2 Associative Property
\nof Addition (a + b) + c = a + (b + c)
\nof Multiplication (a \u00d7 b) \u00d7 c = a \u00d7 (b \u00d7 c)
\n3 Distributive Property
\nacross Addition a(b + c) = a(b) + a(c)
\nacross Subtraction a(b – c) = a(b) – a(c)<\/p>\n

Do You Understand?<\/strong><\/p>\n

Question 1.
\n? Essential Question<\/strong> How can you identify and write equivalent expressions?
\nAnswer:
\nCombine any like terms on each side of the equation x-terms with x-terms and constants with constants. Arrange the terms in the same order, usually x-term before constants. If all of the terms in the two expressions are identical, then the two expressions are equivalent.<\/p>\n

Question 2.
\nUse Structure<\/strong> which property of operations could you use to write an equivalent expression for y + \\(\\frac{1}{2}\\)? Write the equivalent expression.
\nAnswer:
\nThe given expression is: y + \\(\\frac{1}{2}\\)
\nNow,
\nTo write the equivalent expression for the given expression,
\nWe will use “Distributive Property of Addition”
\nNow,
\ny + \\(\\frac{1}{2}\\)
\n= \\(\\frac{2 (y) + 1}{2}\\)
\n= \\(\\frac{2y + 1}{2}\\)
\nSo,
\nThe equivalent expression for the given expression is: \\(\\frac{2y + 1}{2}\\)
\nHence, from the above,
\nWe can conclude that
\nThe property of operation you could use to write y + \\(\\frac{1}{2}\\) is: Distributive Property of Addition
\nThe equivalent expression for y + \\(\\frac{1}{2}\\) is: \\(\\frac{2y + 1}{2}\\)<\/p>\n

Question 3.
\nGeneralize<\/strong> Are z3<\/sup> and 3z equivalent expressions? Explain.
\nAnswer:
\nThe given expressions are: z\u00b3 and 3z
\nNow,
\nz\u00b3 = z \u00d7 z \u00d7 z
\n3z = z + z + z
\nHence, from the above,
\nWe can conclude that
\nz\u00b3 and 3z are not equivalent expressions<\/p>\n

Question 4.
\nAre the expressions 3(y + 1) and 3y + 3 equivalent for y = 1? y = 2? y = 3?
\nAnswer:
\nThe given expressions are: 3 (y + 1) and 3y + 3
\nNow,
\nFor y = 1,
\n3 (y + 1) = 3 (1 + 1)
\n= 3 (2) = 6
\n3y + 3 = 3 (1) + 3
\n= 3 + 3 = 6
\nNow,
\nFor y = 2,
\n3 (y + 1) = 3 (2 + 1)
\n= 3 (3) = 9
\n3y + 3 = 3 (2) + 3
\n= 6 + 3 = 9
\nNow,
\nFor y = 3,
\n3 (y + 1) = 3 (3 + 1)
\n= 3 (4) = 12
\n3y + 3 = 3 (3) + 3
\n= 9 + 3 = 12
\nSo,
\nFor y = 1, y = 2, and y = 3
\n3 (y + 1) = 3y + 3
\nHence, from the above,
\nWe can conclude that
\n3 (y + 1) is equivalent to 3y + 3 for y = 1, y = 2, and y = 3<\/p>\n

Question 5.
\nConstruct Arguments<\/strong> Are the expressions 3(y + 1) and 3y + 3 equivalent for any value of y? Explain.
\nAnswer:
\nThe given expressions are: 3 (y + 1) and 3y + 3
\nNow,
\n3 (y + 1) = 3 (y) + 3 (1)
\n= 3y + 3
\nSo
\n3 (y + 1) = 3y + 3
\nWhere,
\ny will be an integer or a whole number
\nHence, from the above,
\nWe can conclude that
\nThe expressions 3 (y + 1) and 3y + 3 are equivalent for any value of y<\/p>\n

Do You Know How?<\/strong><\/p>\n

In 6-8, use properties of operations to complete the equivalent expressions.<\/strong><\/p>\n

Question 6.
\n\"Envision
\nAnswer:
\nThe given expression is: 2 (r + 3)
\nNow,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe equivalent expression for the given expression is: 2r + 6<\/p>\n

Question 7.
\n\"Envision
\nAnswer:
\nThe given expression is: 6 (4s – 1)
\nNow,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe equivalent expression for the given expression is 24s – 6<\/p>\n

Question 8.
\n\"Envision
\nAnswer:
\nThe given expression is: 8t + 2
\nNow,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe equivalent expression for the given expression is: 2 (4t + 1)<\/p>\n

Question 9.
\nComplete the table below.
\n\"Envision
\nAnswer:
\nThe given expressions are:
\na. 12x – 6\u00a0 \u00a0 b. 3x + 3\u00a0 \u00a0 c. 6 (2x – 1)
\nNow,
\na.
\nFor x = 1,
\n12x – 6 = 12 (1) – 6
\n= 12 – 6
\n= 6
\nFor x = 2,
\n12x – 6 = 12 (2) – 6
\n= 24 – 6
\n= 18
\nFor x = 3,
\n12x – 6 = 12 (3) – 6
\n= 36 – 6
\n= 30
\nb.
\nFor x = 1,
\n3x + 3 = 3 (1) + 3
\n= 3 + 3
\n= 6
\nFor x = 2,
\n3x + 3 = 3 (2) + 3
\n= 6 + 3
\n= 9
\nFor x = 3,
\n3x + 3 = 3 (3) + 3
\n= 9 + 3
\n= 12
\nc.
\nFor x = 1,
\n6 (2x – 1) = 6 [ 2 (1) – 1]
\n= 6
\nFor x = 2,
\n6 (2x – 1) = 6 [2 (2) – 1]
\n= 18
\nFor x = 3,
\n6 (2x – 1) = 6 [2 (3) – 1]
\n= 30
\nHence, from the above,
\nWe can conclude that
\nThe completed table is:
\n\"\"<\/p>\n

Question 10.
\nIn exercise 9, which expressions in the table are equivalent?
\nAnswer:
\nFrom Exercise 9,
\nThe completed table is:
\n\"\"
\nNow,
\nFrom the given table,
\nWe can observe that
\nThe values are the same for 12x – 6 and 6 (2x – 1) i.e.,
\n12x – 6 = 6 (2x – 1)
\nHence, from the above,
\nWe can conclude that
\n12x – 6 and 6 (2x – 1) are equivalent expressions<\/p>\n

Practice & Problem Solving<\/strong><\/p>\n

Scan for Multimedia<\/strong><\/p>\n

Leveled Practice<\/strong>
\nIn 11-20, write equivalent expressions.<\/strong><\/p>\n

Question 11.
\n\"Envision
\nAnswer:
\nThe given expression is: 3 (m + 3)
\nNow,
\n3 (m + 3) = 3 (m) + 3 (3)
\n= 3m + 9
\nHence, from the above,
\nWe can conclude that
\nThe equivalent expression for the given expression is: 3m + 9<\/p>\n

Question 12.
\n\"Envision
\nAnswer:
\nThe given expression is: 20n – 4m
\nNow,
\n20n – 4m = 4 (5n) – 4 (m)
\n= 4 (5n – m)
\nHence, from the above,
\nWe can conclude that
\nThe equivalent expression for the given expression is: 4 (5n – m)<\/p>\n

Question 13.
\n3(x \u2013 6)
\nAnswer:
\nThe given expression is: 3 (x – 6)
\nNow,
\n3 (x – 6) = 3 (x) – 3 (6)
\n= 3x – 18
\nHence, from the above,
\nWe can conclude that
\nThe equivalent expression for the given expression is: 3x – 18<\/p>\n

Question 14.
\n2x + 10
\nAnswer:
\nThe given expression is: 2x + 10
\nNow,
\n2x + 10 = 2 (x) + 2 (5)
\n= 2 (x\u00a0 + 5)
\nHence, from the above,
\nWe can conclude that
\nThe equivalent expression for the given expression is: 2 (x + 5)<\/p>\n

Question 15.
\n8(2y + \\(\\frac{1}{4}\\))
\nAnswer:
\nThe given expression is: 8 (2y + \\(\\frac{1}{4}\\))
\nNow,
\n8 (2y + \\(\\frac{1}{4}\\)) = 8 (2y) + 8 (\\(\\frac{1}{4}\\))
\n= 16y + 2
\nHence, from the above,
\nWe can conclude that
\nThe equivalent expression for the given expression is: 16y + 2<\/p>\n

Question 16.
\n5.7 + (3z + 0.3)
\nAnswer:
\nThe given expression is: 5.7 + (3z + 0.3)
\nNow,
\n5.7 + (3z + 0.3) = 3z + (5.7 + 0.3)
\n= 3z + 6
\nHence, from the above,
\nWe can conclude that
\nThe equivalent expression for the given expression is: 3z + 6<\/p>\n

Question 17.
\n5w – 15
\nAnswer:
\nThe give expression is: 5w – 15
\nNow,
\n5w – 15 = 5 (w) – 5 (3)
\n= 5 (w – 3)
\nHence, from the above,
\nWe can conclude that
\nThe equivalent expression for the given expression is: 5 (w – 3)<\/p>\n

Question 18.
\n2x + 4y
\nAnswer:
\nThe given expression is: 2x + 4y
\nNow,
\n2x + 4y = 2 (x) + 2 (2y)
\n= 2 (x + 2y)
\nHence, from the above,
\nWe can conclude that
\nThe equivalent expression for the given expression is: 2 (x + 2y)<\/p>\n

Question 19.
\n10(y2<\/sup> + 2.45)
\nAnswer:
\nThe given expression is: 10 (y\u00b2 + 2.45)
\nNow,
\n10 (y\u00b2 + 2.45) = 10 (y\u00b2) + 10 (2.45)
\n= 10y\u00b2 + 24.5
\nHence, from the above,
\nWe can conclude that
\nThe equivalent expression for the given expression is: 10y\u00b2 + 24.5<\/p>\n

Question 20.
\n\\(\\frac{3}{4}\\) \u2022 (z3<\/sup>\u20224)
\nAnswer:
\nThe given expression is: \\(\\frac{3}{4}\\) \u00d7 (z\u00b3 \u00d7 4)
\nNow,
\n\\(\\frac{3}{4}\\) \u00d7 (z\u00b3 \u00d7 4) = z\u00b3 \u00d7 \\(\\frac{3}{4}\\) \u00d7 4
\n= 3z\u00b3
\nHence, from the above,
\nWe can conclude that
\nThe equivalent expression for the given expression is: 3z\u00b3<\/p>\n

In 21-24, write the letters of the expressions that are equivalent to the given expression.<\/strong><\/p>\n

Question 21.
\n5(2x + 3)
\na. 10x + 15
\nb. 5x + 15 + 5x
\nc. 10x + 8
\nAnswer:
\nThe given expression is: 5 (2x + 3)
\nNow,
\n5 (2x + 3) = 5 (2x) + 5 (3)
\n= 10x + 15
\nHence, from the above,
\nWe can conclude that
\nThe expression that is equivalent to the given expression is:
\n\"\"<\/p>\n

Question 22.
\n4x – 8
\na. 2(2x – 6)
\nb. 2(2x – 4)
\nc. x – 8 + 3x
\nAnswer:
\nThe given expression is: 4x – 8
\nNow,
\n4x – 8 = 4 (x – 2)
\n= 3x + x – 8
\n= 2 (2x – 4)
\nHence, from the above,
\nWe can conclude that
\nThe expressions that are equivalent to the given expression is:
\n\"\"<\/p>\n

Question 23.
\n12x \u2013 16
\na. 9.6x \u2013 16+ 2.4x
\nb. 3(3x \u2013 5)
\nc. 4(3x \u2013 4)
\nAnswer:
\nThe given expression is: 12x – 16
\nNow,
\n12x – 16 = 4 (3x – 4)
\n= 9.6x + 2.4x – 16
\nHence, from the above,
\nWe can conclude that
\nThe expressions that are equivalent to the given expression is:
\n\"\"<\/p>\n

Question 24.
\n2(6x + \\(\\frac{1}{2}\\))
\na. 12x + 2
\nb. 12x + 1
\nc. 6x + \\(\\frac{1}{2}\\) + 6x + \\(\\frac{1}{2}\\)
\nAnswer:
\nThe given expression is: 2 (6x + \\(\\frac{1}{2}\\))
\nNow,
\n2 (6x + \\(\\frac{1}{2}\\))
\n= 6x + 6x + \\(\\frac{1}{2}\\) + \\(\\frac{1}{2}\\)
\n= 12x + 1
\nHence, from the above,
\nWe can conclude that
\nThe expressions that are equivalent to the given expression is:
\n\"\"<\/p>\n

In 25-27, use the signs at the right.<\/strong><\/p>\n

Question 25.
\nWrite an algebraic expression that represents each purchase.
\na. Mr. Tonkery bought x number of soccer balls and 3 baseballs.
\n\"Envision<\/p>\n

Answer:
\nIt is given that
\nMr. Tonkery bought x number of soccer balls and 3 baseballs.
\nNow,
\nThe given figure is:
\n\"Envision
\nSo,
\nThe total purchase made by Mr. Tonkery = (The number of soccer balls) \u00d7 (The cost of each soccer ball) + (The number of baseballs) \u00d7 (The cost of each baseball)
\n= (x \u00d7 $15) + (3 \u00d7 $6)
\n= $15x + $18
\nHence, from the above,
\nWe can conclude that
\nThe algebraic expression that represents Mr. Tonkery’s purchase is: $15x + $18<\/p>\n

b. Dennis, Eddie, and Felix are on a baseball team. They each bought a baseball and x pairs of sweat socks.
\nAnswer:
\nIt is given that
\nDennis, Eddie, and Felix are on a baseball team. They each bought a baseball and x pairs of sweat socks.
\nNow,
\nThe given figure is:
\n\"Envision
\nSo,
\nThe total purchase made by Dennis, Eddie, and Felix = (The number of baseballs purchased by the three of them) \u00d7 (The cost of each baseball) + (The number of sweat socs purchased by the three of them) \u00d7 (The cost of each sweat socks)
\n= (3 \u00d7 $6) + (3x \u00d7 $5)
\n= $15x + $18
\nHence, from the above,
\nWe can conclude that
\nThe algebraic expression that represents the total purchase made by Dennis, Eddie, and Felix is: $15x + $18<\/p>\n

Question 26.
\nMake Sense and Persevere<\/strong> Suppose x has the same value in both of the expressions you wrote for exercise 25. Are the two expressions you wrote equivalent? Explain.
\nAnswer:
\nIt is given that
\nx has the same value in both of the expressions you wrote for exercise 25.
\nSo,
\nFrom Part (a) and part (b) of Exercise 25,
\n$15x + $18 = $15x + $18
\nHence, from the above,
\nWe can conclude that
\nThe two expressions in exercise 25’s part (a) and part (b) are equivalents<\/p>\n

Question 27.
\nCritique Reasoning<\/strong> Wendy says that soccer balls cost 2\\(\\frac{1}{2}\\) times as much as baseballs. Do you agree? Explain.
\nAnswer:
\nIt is given that
\nWendy says that soccer balls cost 2\\(\\frac{1}{2}\\) times as much as baseballs.
\nNow,
\nThe given figure is:
\n\"Envision
\nNow,
\nFrom the given figure,
\nWe can observe that
\nThe cost of each soccer ball is: $15
\nThe cost of each baseball is: $6
\nNow,
\nLet the number of times as soccer balls as many as base balls be x
\nSo,
\nx \u00d7 $6 = $15
\nx = \\(\\frac{15}{6}\\)
\nx = \\(\\frac{5}{2}\\)
\nx = 2\\(\\frac{1}{2}\\)
\nHence, from the above,
\nWe can conclude that
\nWe can agree with Wendy<\/p>\n

Question 28.
\nUse Structure<\/strong> Write an algebraic expression to represent the area of the rectangular rug. Then use properties of operations to write an equivalent expression.
\n\"Envision
\nAnswer:
\nThe given figure is:
\n\"Envision
\nNow,
\nFrom the above figure,
\nWe can observe that
\nThe length of the rectangular rug is: 2 (x – 1)
\nThe width of the rectangular rug is: 5
\nNow,
\nWe know that,
\nThe area of a rectangle = Length \u00d7 Width
\nSo,
\nThe area of a rectangular rug = 2 (x – 1) \u00d7 5
\n= [2 (x) – 2 (1)] \u00d7 5
\n= (2x – 2) \u00d7 5
\n= 5 (2x) – 5 (2)
\n= 10x – 10
\nHence from the above,
\nWe can conclude that
\nThe algebraic expression that represents the area of the given rectangular rug is: 10x – 10<\/p>\n

Question 29.
\nCritique Reasoning<\/strong> Jamie says that the expressions 6x – 2x + 4 and 4(x + 1) are not equivalent because one expression has a term that is subtracted and the other does not. Do you agree? Explain.
\nAnswer:
\nIt is given that
\nJamie says that the expressions 6x – 2x + 4 and 4(x + 1) are not equivalent because one expression has a term that is subtracted and the other does not.
\nNow,
\nThe given expressions are: 6x – 2x + 4 and 4 (x + 1)
\nNow,
\n6x – 2x + 4 = 4x + 4
\nNow,
\n4 (x + 1) = 4 (x) + 4 (1)
\n= 4x + 4
\nSo,
\n6x – 2x + 4 = 4x + 4
\nHence, from the above,
\nWe can conclude that
\nWe can not agree with Jamie<\/p>\n

Question 30.
\nAre the two expressions shown below equivalent? Explain.
\n4(n + 3) \u2013 (3 + n) and 3n + 9
\nAnswer:
\nThe given expressions are: 4 (n + 3) – (3 + n) and 3n + 9
\nNow,
\n4 (n + 3) – (3 + n) = 4 (n) + 4 (3) – 3 – n
\n= 4n + 12 – 3 – n
\n= 3n + 9
\nSo,
\n4 (n + 3) – (3 + n) = 3n + 9
\nHence, from the above,
\nWe can conclude that
\nThe given two expressions are equivalent expressions<\/p>\n

Question 31.
\nCritique Reasoning<\/strong> Chris says that the expression 4n – 2 can be written as 2(2n – 1). Do you agree? Explain.
\nAnswer:
\nIt is given that
\nChris says that the expression 4n – 2 can be written as 2(2n – 1)
\nNow,
\nThe given expression is: 4n – 2
\nNow,
\n4n – 2 = 2 (2n – 1)
\nHence, from the above,
\nWe can conclude that
\nWe can agree with Chris<\/p>\n

Question 32.
\nHigher Order Thinking<\/strong> Write an expression that has only one term and is equivalent to the expression below.
\n(f \u2022 g2<\/sup>) + 5 \u2013 (g2 <\/sup>\u2022 f)
\nAnswer:
\nThe given expression is: (f \u00d7 g\u00b2) + 5 – (g\u00b2 \u00d7f )
\nNow,
\nWe know that,
\nAccording to the Associative Property of Multiplication,
\na \u00d7 b = b \u00d7 a
\nSo,
\nf \u00d7 g\u00b2 = g\u00b2 \u00d7 f
\nSo,
\n(f \u00d7 g\u00b2) + 5 – (g\u00b2 \u00d7f ) = 0 + 5
\n= 5
\nHence, from the above,
\nWe can conclude that
\nThe expression that has only one term and is equivalent to the given expression is: 5<\/p>\n

Question 33.
\nConstruct Arguments<\/strong> A Florida college golf team with 14 members is planning an awards banquet. To find the total cost of the meals, the team uses the expression 5(g + 14), where g is the number of guests attending the banquet. A team member says that an equivalent expression is 5g + 14. Do you agree? Explain.
\n\"Envision
\nAnswer:
\nIt is given that
\nA Florida college golf team with 14 members is planning an awards banquet. To find the total cost of the meals, the team uses the expression 5(g + 14), where g is the number of guests attending the banquet. A team member says that an equivalent expression is 5g + 14
\nNow,
\nThe given figure is:
\n\"Envision
\nNow,
\nThe given expression is:
\n5 (g + 14) = 5 (g) + 5 (14)
\n= 5g + 70
\nSo,
\n5 (g + 14) \u2260 5g + 14
\nHence, from the above,
\nWe can conclude that
\nWe can not agree that the equivalent expression of 5 (g + 14) is 5g + 14<\/p>\n

Assessment Practice<\/strong><\/p>\n

Question 34.
\nSelect each expression that is equivalent to 8.5 + (2s + 0.5).
\n\"Envision (8.5 + 2s) + 0.5
\n\"Envision (8.5 + 0.5) + 2s
\n\"Envision 9 + 2
\n\"Envision 2(4.5 + s)
\n\"Envision 8.5(2s + 0.5)
\nAnswer:
\nThe given epression is: 8.5 + (2s + 0.5)
\nHence, from the above,
\nWe can conclude that
\nThe expressions that are equivalent to 8.5 + (2s + 0.5) are:
\n\"\"<\/p>\n

Question 35.
\nSelect each expression that is equivalent to 5(n + 4).
\n\"Envision 5n + 4
\n\"Envision 5n + 20
\n\"Envision 15 + 5n + 5
\n\"Envision 5(n + 3) + 5
\n\"Envision 5n + 54
\nAnswer:
\nThe given expression is: 5 (n + 4)
\nHence, from the above,
\nWe can conclude that
\nThe expressions that are equivalent to 5 (n + 4) are:
\n\"\"<\/p>\n

Lesson 3.7 Simplify Algebraic expressions<\/h3>\n

Solve & Discuss It!<\/strong><\/p>\n

Write an expression equivalent to x + 5 + 2x + 2.
\n\"Envision
\nMake Sense and Persevere<\/strong> Use what you know about algebraic expressions and properties of operations to make sense of the problem.
\nAnswer:
\nThe given expression is: x + 5 + 2x + 2
\nNow,
\nWe know that,
\nThe terms that have the same variable are called “Like terms” and we use the Order of operations to write the equivalent expression
\nSo,
\nx + 5 + 2x + 2 = (x + 2x) + (5 + 2)
\n= 3x + 7
\nHence, from the above,
\nWe can conclude that
\nThe equivalent expression for the given expression is: 3x + 7<\/p>\n

Focus on math practices<\/strong>
\nBe Precise<\/strong> How do you know that the expression you wrote is equivalent to x + 5 + 2x + 2?
\nAnswer:
\nThe given expression is: x + 5 + 2x + 2
\nNow,
\nWe know that,
\nThe terms that have the same variable are called “Like terms” and we use the Order of operations to write the equivalent expression
\nHence, from the above,
\nWe can conclude that
\nWe will use the Order of operations and “Like terms” to solve the given expression for its equivalent expression<\/p>\n

VISUAL LEARNING<\/strong><\/p>\n

? Essential Question<\/strong> How can you simplify algebraic expressions?
\nAnswer:
\nThe steps to simplify the algebraic expressions are:
\na. When simplifying mathematical expressions perform the operations in the following order: Parentheses and other Grouping Symbols.
\nb. Exponents.
\nc. Multiplication and Division.
\nd. Addition and Subtraction.<\/p>\n

Try It!<\/strong>
\nSimplify the expression 4z \u2013 z + z \u2013 2z.<\/p>\n

4z \u2013 z + z – 2z
\n\"Envision
\nAnswer:
\nThe given expression is: 4z – z + z – 2z
\nNow,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: 2z<\/p>\n

Convince Me!<\/strong> How do you know that the expression 2x + 4y is not equivalent to 6xy?
\nAnswer:
\nThe given expressions are: 2x + 4y and 6xy
\nNow,
\n2x + 4y = 2 (x + 2y)
\nNow,
\nWhen we observe the two expressions,
\n2x + 4y contains “Addition” operation
\n6xy contains only the “Product” operation
\nSo,
\n2x + 4y \u2260 6xy
\nHence, from the above,
\nWe can conclude that
\n2x + 4y is not equivalent to 6xy with the help of arithmetic symbols<\/p>\n

Try It!<\/strong>
\nPark rangers add another section to the trail, represented by the expression in \\(\\frac{1}{2}\\)n + n + \\(\\frac{1}{2}\\). Write an expression for the new total length of the trail. Then write a simplified equivalent expression.
\nAnswer:
\nIt is given that
\nPark rangers add another section to the trail, represented by the expression in \\(\\frac{1}{2}\\)n + n + \\(\\frac{1}{2}\\)
\nNow,
\nThe given figure is:
\n\"\"
\nSo,
\nThe new total length of the trail = (The old total length of the trail that is present in Example 2) + \\(\\frac{1}{2}\\) + n + \\(\\frac{1}{2}\\)
\n= 3\\(\\frac{2}{3}\\)n + 4 + \\(\\frac{1}{2}\\)n + n + \\(\\frac{1}{2}\\)
\n= \\(\\frac{11}{3}\\)n + 4 + \\(\\frac{1}{2}\\)n + n + \\(\\frac{1}{2}\\)
\n= \\(\\frac{11}{3}\\)n + \\(\\frac{3}{2}\\)n + \\(\\frac{9}{2}\\)
\n= \\(\\frac{31}{6}\\)n + \\(\\frac{9}{2}\\)
\nHence, from the above,
\nWe can conclude that
\nThe simplified expression of the new total length of the trail is: \\(\\frac{31}{6}\\)n + \\(\\frac{9}{2}\\)<\/p>\n

Try It!<\/strong>
\nSuppose Vanna increases her rate by 3.5 times and her expenses also increase by 3.5 times this summer. Write two equivalent expressions to represent how much she can earn mowing and raking grass.
\nAnswer:
\nIt is given that
\nVanna increases her rate by 3.5 times and her expenses also increase by 3.5 times this summer.
\nNow,
\nThe given figure is:
\n\"\"
\nSo,
\nThe algebraic expression that represents the amount of money Vanna can earn from mowing and raking grass = 3.5 ($20.50x + $5.50x – $10)
\nNow,
\nBy using the Distributive Property,
\n3.5 ($20.50x + $5.50x – $10)
\n= 3.5 ($26x – $10)
\n= 3.5 ($26x) – 3.5 ($10)
\n= $91x – $35
\n= 7 ($13x – $5)
\nHence, from the above,
\nWe can conclude that
\nThe two equivalent expressions that represent the amount of money Vanna earned from mowing and raking grass is:
\n$91x – $35 (or) 7 ($13x – $5)<\/p>\n

KEY CONCEPT<\/strong>
\nYou can combine like terms to write equivalent expressions. Like terms have the same variable part.
\n2x + 6 + 5x + 4 ……… Identify like terms
\n= 2x + 5x + 6 + 4 ………… Commutative Property of Addition
\n= 7x + 10
\n2x + 6 + 5x + 4 = 7x + 10<\/p>\n

Do You Understand?<\/strong><\/p>\n

Question 1.
\n? Essential Question<\/strong> How can you simplify algebraic expressions?
\nAnswer:
\nThe steps to simplify the algebraic expressions are:
\na. When simplifying mathematical expressions perform the operations in the following order: Parentheses and other Grouping Symbols.
\nb. Exponents.
\nc. Multiplication and Division.
\nd. Addition and Subtraction.<\/p>\n

Question 2.
\nExplain how you know which terms to combine when combining like terms.
\nAnswer:
\nThe terms that have identical variable parts (same variable(s) and same exponent(s)) are called “Like terms”. When simplifying using addition and subtraction, you combine \u201clike terms\u201d by keeping the “<\/b>like term” and adding or subtracting the numerical coefficients<\/p>\n

Question 3.
\nConstruct Arguments<\/strong> Explain why the expression 2y – y can be written as y.
\nAnswer:
\nThe given expression is: 2y – y
\nNow,
\nBy using the Order of operations,
\n2y – y = 2y – 1y
\n= y (2 – 1)
\n= y (1)
\n= y
\nHence, from the above,
\nWe can conclude that
\n2y – y can be written as y by using the Order of operations<\/p>\n

Question 4.
\nExplain why the expressions \\(\\frac{1}{2}\\)x + \\(\\frac{1}{2}\\)x and x are equivalent.
\nAnswer:
\nThe given expression is: \\(\\frac{1}{2}\\)x + \\(\\frac{1}{2}\\)x
\nNow,
\nBy using the order of Operations,
\n\\(\\frac{1}{2}\\)x + \\(\\frac{1}{2}\\)x
\n= \\(\\frac{x}{2}\\) + \\(\\frac{x}{2}\\)
\n= \\(\\frac{x + x}{2}\\)
\n= \\(\\frac{2x}{2}\\)
\n= \\(\\frac{x}{1}\\)
\n= x
\nSo,
\n\\(\\frac{1}{2}\\)x + \\(\\frac{1}{2}\\)x = x
\nHence, from the above,
\nWe can concldue that
\n\\(\\frac{1}{2}\\)x + \\(\\frac{1}{2}\\)x and x are equivalents by using the properties of addition in fraction and order of operations<\/p>\n

Question 5.
\nCritique Reasoning<\/strong> Henry wrote 4z2<\/sup> – z2<\/sup> as 4. Are 4z2<\/sup> – z2<\/sup> and 4 equivalent expressions? Explain.
\nAnswer:
\nThe given expression is: 4z\u00b2 – z\u00b2
\nNow,
\nBy using the order of operations,
\n4z2<\/sup> – z2<\/sup>
\n= 4z2<\/sup> – 1z2<\/sup>
\n= z2<\/sup> (4 – 1)
\n= z2<\/sup> (3)
\n= 3z2<\/sup>
\nSo,
\n4z2<\/sup> – z2<\/sup>\u00a0 \u2260 4
\nHence, from the above,
\nWe can conclude that
\n4z2<\/sup> – z2<\/sup> is not equivalent to 4<\/p>\n

Do You Know How?<\/strong><\/p>\n

In 6-15, simplify each expression.<\/strong><\/p>\n

Question 6.
\nx + x + x + x
\nAnswer:
\nThe given expression is: x + x + x + x
\nNow,
\nBy using the Order of Operations,
\nx + x + x + x
\n= 1x + 1x + 1x + 1x
\n= x (1 + 1 + 1 + 1)
\n= x (4)
\n= 4x
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: 4x<\/p>\n

Question 7.
\n4y – y
\nAnswer:
\nThe given expression is: 4y – y
\nNow,
\nBy using the Order of Operations,
\n4y – y
\n= 4y – 1y
\n= y (4 – 1)
\n= y (3)
\n= 3y
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: 3y<\/p>\n

Question 8.
\n7y – 4.5 – 6y
\nAnswer:
\nThe given expression is: 7y – 4.5 – 6y
\nNow,
\nBy using the Order of Operations,
\n7y – 4.5 – 6y
\n= (7y – 6y) – 4.5
\n= y – 4.5
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: y – 4.5<\/p>\n

Question 9.
\n4x + 2 – \\(\\frac{1}{2}\\)x
\nAnswer:
\nThe given expression is: 4x + 2 – \\(\\frac{1}{2}\\)x
\nNow,
\nBy using the Order of Operations,
\n4x + 2 – \\(\\frac{1}{2}\\)x
\n= (4x – \\(\\frac{1}{2}\\)x) + 2
\n= \\(\\frac{8x – x}{2}\\) + 2
\n= \\(\\frac{7x}{2}\\) + 2
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: \\(\\frac{7x}{2}\\) + 2<\/p>\n

Question 10.
\n3 + 3y – 1 + y
\nAnswer:
\nThe given expression is: 3 + 3y – 1 + y
\nNow,
\nBy using the Order of Operations,
\n3 + 3y – 1 + y
\n= (3 – 1) + (3y + y)
\n= 2 + 4y
\n= 4y + 2
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: 4y + 2<\/p>\n

Question 11.
\nx + 6x
\nAnswer:
\nThe given expression is: x + 6x
\nNow,
\nBy using the Order of Operations,
\nx + 6x
\n= 1x + 6x
\n= x (1 + 6)
\n= x (7)
\n= 7x
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: 7x<\/p>\n

Question 12.
\n0.5w + 1.7w \u2013 0.5
\nAnswer:
\nThe given expression is: 0.5w + 1.7w – 0.5
\nNow,
\nBy using the Order of Operations,
\n0.5w + 1.7w – 0.5
\n= (0.5w + 1.7w) – 0.5
\n= w (0.5 + 1.7) – 0.5
\n= 2.2w – 0.5
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: 2.2w – 0.5<\/p>\n

Question 13.
\n12\\(\\frac{1}{3}\\)b + 6\\(\\frac{2}{3}\\) – 10\\(\\frac{2}{3}\\)b
\nAnswer:
\nThe given expression is:
\n12\\(\\frac{1}{3}\\)b + 6\\(\\frac{2}{3}\\) – 10\\(\\frac{2}{3}\\)b
\nNow,
\nConvert the given mixed numbers into fractions
\nSo,
\nThe given expression is: \\(\\frac{37}{3}\\)b + \\(\\frac{20}{3}\\) – \\(\\frac{32}{3}\\)b
\nNow,
\nBy using the Order of Operations,
\n\\(\\frac{37}{3}\\)b + \\(\\frac{20}{3}\\) – \\(\\frac{32}{3}\\)b
\n= (\\(\\frac{37}{3}\\)b – \\(\\frac{32}{3}\\)b) + \\(\\frac{20}{3}\\)
\n= b (\\(\\frac{37}{3}\\) – \\(\\frac{32}{3}\\)) + \\(\\frac{20}{3}\\)
\n= \\(\\frac{5}{3}\\)b + \\(\\frac{20}{3}\\)
\n= \\(\\frac{1}{3}\\) (5b + 20)
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: \\(\\frac{1}{3}\\) (5b + 20)<\/p>\n

Question 14.
\n\\(\\frac{3}{4}\\)x + 2 + 3x – \\(\\frac{1}{2}\\)
\nAnswer:
\nThe given expression is: \\(\\frac{3}{4}\\)x + 2 + 3x – \\(\\frac{1}{2}\\)
\nNow,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: \\(\\frac{15}{4}\\)x + \\(\\frac{3}{2}\\)<\/p>\n

Question 15.
\n3.2x + 6.5 – 2.4x – 4.4
\nAnswer:
\nThe given expression is: 3.2x + 6.5 – 2.4x – 4.4
\nNow,
\nBy using the Order of Operations,
\n3.2x + 6.5 – 2.4x – 4.4
\n= (3.2x – 2.4x) + (6.5 – 4.4)
\n= x (3.2 – 2.4) + 2.1
\n= x (0.8) + 2.1
\n= 0.8x + 2.1
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: 0.8x + 2.1<\/p>\n

Practice & Problem Solving<\/strong><\/p>\n

Scan for Multimedia<\/strong><\/p>\n

Leveled Practice<\/strong>
\nIn 16-26, combine like terms to simplify each expression.<\/strong><\/p>\n

Question 16.
\n\"Envision
\nAnswer:
\nThe given expression is: 2.1x\u00b2 + 3 – 0.5x\u00b2 – 1
\nNow,
\n2.1x\u00b2 + 3 – 0.5x\u00b2 – 1
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: \\(\\frac{8}{5}\\)x\u00b2 + 2<\/p>\n

Question 17.
\n\"Envision
\nAnswer:
\nThe given expression is: \\(\\frac{2}{3}\\)n + 6 +3n – \\(\\frac{2}{3}\\)
\nNow,
\n\\(\\frac{2}{3}\\)n + 6 +3n – \\(\\frac{2}{3}\\)
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: \\(\\frac{11}{3}\\)n + \\(\\frac{16}{3}\\)<\/p>\n

Question 18.
\n5 + 3w + 3 – w
\nAnswer:
\nThe given expression is: 5 + 3w + 3 – w
\nNow,
\n5 + 3w + 3 – w
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: 2w + 8<\/p>\n

Question 19.
\n5w – 5w
\nAnswer:
\nThe given expression is: 5w – 5w
\nNow,
\n5w – 5w
\n= w (5 – 5)
\n= w (0)
\n= 0
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: 0<\/p>\n

Question 20.
\n2x + 5 + 3x + 6
\nAnswer:
\nThe givene xpression is: 2x + 5 + 3x + 6
\nNow,
\n2x + 5 + 3x + 6
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: 5x + 11<\/p>\n

Question 21.
\n\\(\\frac{3}{4}\\)z3<\/sup> + 4 – \\(\\frac{1}{4}\\)z3<\/sup>
\nAnswer:
\nThe given expression is: \\(\\frac{3}{4}\\)z3<\/sup> + 4 – \\(\\frac{1}{4}\\)z3<\/sup>
\nNow,
\n\\(\\frac{3}{4}\\)z3<\/sup> + 4 – \\(\\frac{1}{4}\\)z3<\/sup>
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: \\(\\frac{1}{2}\\)z\u00b3 + 4<\/p>\n

Question 22.
\n3.4m + 2.4m
\nAnswer:
\nThe given expression is: 3.4m + 2.4m
\nNow,
\n3.4m + 2.4m
\n= m (3.4 + 2.4)
\n= m (5.8)
\n= 5.8m
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: 5.8m<\/p>\n

Question 23.
\n4.2n + 5 – 3.2n
\nAnswer:
\nThe givene xpression is: 4.2n + 5 – 3.2n
\nNow,
\n4.2n + 5 – 3.2n
\n= (4.2n – 3.2n) + 5
\n= n (4.2 – 3.2) + 5
\n= n (1) + 5
\n= n + 5
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: n + 5<\/p>\n

Question 24.
\nq5<\/sup> + q5<\/sup> + q5<\/sup>
\nAnswer:
\nThe given expression is: q5<\/sup> + q5<\/sup> + q5<\/sup>
\nNow,
\nq5<\/sup> + q5<\/sup> + q5<\/sup>
\n= q5<\/sup> (1 + 1 + 1)
\n= q5<\/sup> (3)
\n= 3q5<\/sup>
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: 3q5<\/sup><\/p>\n

Question 25.
\n3x + \\(\\frac{1}{4}\\) + 2y + \\(\\frac{1}{4}\\) + 7x – y
\nAnswer:
\nThe given expression is: 3x + \\(\\frac{1}{4}\\) + 2y + \\(\\frac{1}{4}\\) + 7x – y
\nNow,
\n3x + \\(\\frac{1}{4}\\) + 2y + \\(\\frac{1}{4}\\) + 7x – y
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: 10x + y + \\(\\frac{1}{2}\\)<\/p>\n

Question 26.
\n1.5z2<\/sup> + 4.5 + 6z – 0.3 – 3z + z2<\/sup>
\nAnswer:
\nThe given expression is: 1.5z2<\/sup> + 4.5 + 6z – 0.3 – 3z + z2<\/sup>
\nNow,
\n1.5z2<\/sup> + 4.5 + 6z – 0.3 – 3z + z2<\/sup>
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: \\(\\frac{5}{2}\\)z\u00b2 + 3z + \\(\\frac{21}{5}\\)<\/p>\n

Question 27.
\nUse Structure<\/strong> Use the table at the right. Yolanda is planning a party that will take place in three rooms.
\n\"Envision
\na. Write an expression that can be used to represent the total amount Yolanda will need to rent all three rooms and the sound system for t hours.
\nAnswer:
\nIt is given that
\nYolanda is planning a party that will take place in three rooms.
\nNow,
\nThe given table is:
\n\"Envision
\nSo,
\nThe amount of money Yolanda will need to rent all three rooms and the sound system for t hours =
\n(The total rental fee of the three rooms per hour) \u00d7 (The total number of hours the three rooms are rented) + (The total fee of Sound system)
\n= ($25 + $20 + $50) \u00d7 t + ($15 + $10)
\n= $95 \u00d7 t + $25
\n= $95t + $25
\nHence, from the above,
\nWe can conclude that
\nThe expression that can be used to represent the total amount Yolanda will need to rent all three rooms and the sound system for t hours is: $95t + $25<\/p>\n

b. How can you use a property to write a simplified equivalent expression?
\nAnswer:
\nThe properties used to write a simplified equivalent expression are:
\na. Commutative Property of Addition : When adding, changing the order of the numbers does not change the sum.
\nb. Commutative Property of Multiplication
\nc. Associative Property of Addition
\nd. Associative Property of Multiplication
\ne. Distributive Property<\/p>\n

In 28-30, use the diagram at the right.<\/strong><\/p>\n

Question 28.
\nWrite an algebraic expression for the perimeter of the swimming pool.
\n\"Envision
\nAnswer:
\nThe given figure is:
\n\"Envision
\nNow,
\nFrom the given figure,
\nWe can observe that
\nThe swimming pool is in the form of a rectangle
\nNow,
\nFrom the given figure,
\nWe can observe that
\nThe length of the swimming pool is: 2y + 1
\nThe width of the swimming pool is: y
\nNow,
\nWe know that,
\nThe perimeter of a recatngle = 2 \u00d7 (Length + Width)
\nSo,
\nThe perimeter of the given swimming pool = 2 \u00d7 (Length of the swimming pool + Width of the swimming pool)
\n= 2 \u00d7 (2y + 1 + y)
\n= 2 \u00d7 (3y + 1)
\n= 2 (3y + 1)
\nHence,f rom the above,
\nWe can conclude that
\nThe algebraic expression that represents the perimeter of the swimming pool is: 2 (3y + 1)<\/p>\n

Question 29.
\nUse Structure<\/strong> Write a new expression equivalent to the expression you wrote for exercise 28.
\nAnswer:
\nFrom Exercise 28,
\nWe know that,
\nThe algebraic expression that represents the perimeter of the swimming pool is: 2 (3y + 1)
\nNow,
\nThe given expression is: 2 (3y + 1)
\nNow,
\n2 (3y + 1)
\n= 2 (3y) + 2 (1)
\n= 6y + 2
\nHence, from the above,
\nWe can conclude that
\nThe new expression that is equivalent to the expression you wrote for Exercise 28 is: 6y + 2<\/p>\n

Question 30.
\nJustify that the two expressions are equivalent.
\nAnswer:
\nFrom Exercise 28 and 29
\nWe know that,
\nThe expressions are: 2 (3y + 1) and 6y + 2
\nNow,
\n2 (3y + 1) = 2 (3y) + 2 (1)
\n= 6y + 2
\nSo,
\n2 (3y + 1) = 6y + 2
\nHence, from the above,
\nWe can conclude that
\nThe two expressions that are from Exercise 28 and Exercise 29 are equivalent expressions<\/p>\n

Question 31.
\nRodney rewrote the expression \\(\\frac{1}{2}\\)(2x + 7) as x + 3\\(\\frac{1}{2}\\). Which property of operations did Rodney use?
\nAnswer:
\nIt is given that
\nRodney rewrote the expression \\(\\frac{1}{2}\\)(2x + 7) as x + 3\\(\\frac{1}{2}\\).
\nNow,
\nThe given expression is: \\(\\frac{1}{2}\\) (2x + 7)
\nNow,
\nBy using the Distributive Property,
\n\\(\\frac{1}{2}\\) (2x + 7)
\n= \\(\\frac{1}{2}\\) (2x) + \\(\\frac{1}{2}\\) (7)
\n= \\(\\frac{2x}{2}\\) + \\(\\frac{7}{2}\\)
\n= x + \\(\\frac{7}{2}\\)
\n= x + 3\\(\\frac{1}{2}\\)
\nHence, from the above,
\nWe can conclude that
\nRodney used “Distributive Property” to solve the given expression<\/p>\n

Question 32.
\nConstruct Arguments<\/strong> Annie said that she simplified the expression 6.5(x + 0.5x + 1) by writing the equivalent expression 6.5x + 3.25x + 6.5. Do you agree? Explain.
\nAnswer:
\nIt is given that
\nAnnie said that she simplified the expression 6.5(x + 0.5x + 1) by writing the equivalent expression 6.5x + 3.25x + 6.5
\nNow,
\nThe given expression is: 6.5 (x + 0.5x + 1)
\nNow,
\nBy using the Distributive Property,
\n6.5 (x + 0.5x + 1)
\n= 6.5 (x) + 6.5 (0.5x) + 6.5 (1)
\n= 6.5x + 3.25x + 6.5
\nHence,from the above,
\nWe can conclude that
\nWe can agree with Annie<\/p>\n

Question 33.
\nCritique Reasoning<\/strong> Thea said that the expressions 4x – 3x + 2 and x + 2 are equivalent. Is Thea correct? Explain.
\nAnswer:
\nIt is given that
\nThea said that the expressions 4x – 3x + 2 and x + 2 are equivalent.
\nNow,
\nThe given expression is: 4x – 3x + 2
\nNow,
\n4x – 3x + 2
\n= (4x – 3x) + 2
\n= x + 2
\nSo,
\n4x – 3x + 2 = x + 2
\nHence, from the above,
\nWe can conclude that
\nThea is correct<\/p>\n

Question 34.
\nHigher Order Thinking<\/strong> Write an equivalent expression for the expression shown below.
\n\\(\\frac{a}{3}\\) + \\(\\frac{a}{3}\\) + \\(\\frac{a}{3}\\)
\nAnswer:
\nThe given expression is:
\n\\(\\frac{a}{3}\\) + \\(\\frac{a}{3}\\) + \\(\\frac{a}{3}\\)
\nNow,
\n\\(\\frac{a}{3}\\) + \\(\\frac{a}{3}\\) + \\(\\frac{a}{3}\\)
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: a<\/p>\n

Assessment Practice<\/strong><\/p>\n

Question 35.
\nSelect all expressions that are equivalent to 8x + 3 + 5x \u2013 2x.
\n\"Envision 13x + 3 – 2x
\n\"Envision 11x + 3x
\n\"Envision 11 + 3x
\n\"Envision 14x
\n\"Envision 11x + 3
\nAnswer:
\nThe given expression is: 8x + 3 + 5x – 2x
\nNow,
\nBy using the Order of Operations,
\n8x + 3 + 5x – 2x
\n= (8x + 5x) + 3 – 2x
\n= 13x + 3 – 2x
\n= (13x – 2x) + 3
\n= 11x + 3
\nHence, from the above,
\nWe can conclude that
\nAll expressions that are equivalent to 8x + 3 + 5x \u2013 2x are:
\n\"\"<\/p>\n

Question 36.
\nSelect which expression is equivalent to or NOT equivalent to the given expression.
\n\"Envision
\nAnswer:
\nThe givene xpression is: 2x + 7 + 6x – x
\nNow,
\nBy using the Order of Operations,
\n2x + 7 + 6x – x
\n= (2x + 6x – x) + 7
\n= (8x – x) + 7
\n= 7x + 7
\n= 7 + 7x
\nHence, from the above,
\nWe can conclude that
\nThe expression is equivalent to or not equivalent to the given expression is:
\n\"\"<\/p>\n

Topic 3 Review<\/h3>\n

? Topic Essential Question<\/strong><\/p>\n

What are expressions and how can they be written and evaluated?
\nAnswer:
\nExpressions\u00a0are\u00a0mathematical\u00a0statements that have a minimum of two terms containing numbers or variables, or both, connected by an operator in between.
\nTo evaluate an algebraic expression means\u00a0to find the value of the expression when the variable is replaced by\u00a0a given number. To evaluate an expression, we substitute the given number for the variable in the expression and then simplify the expression using the order of operations.<\/p>\n

Vocabulary Review<\/strong><\/p>\n

Complete each definition with a vocabulary word.<\/strong><\/p>\n

Vocabulary<\/strong>
\nalgebraic expression
\nfactor tree
\ncoefficient
\nlike terms
\nexponent
\nvariable<\/p>\n

Question 1.
\nA(n) ____ tells the number of times the base is used as a factor.
\nAnswer:
\nWe know that,
\nAn “Exponent” tells the number of times the base is used as a factor.
\nHence, from the above,
\nWe can conclude that
\nThe term that is suitable for the given definition is: Exponent<\/p>\n

Question 2.
\nA letter or symbol that represents an unknown quantity is a(n) _____
\nAnswer:
\nWe know that,
\nA letter or symbol that represents an unknown quantity is a “variable”
\nHence, from the above,
\nWe can conclude that
\nThe term that is suitable for the given definition is: Variable<\/p>\n

Question 3.
\nA diagram that shows the prime factors of a composite number is a(n) ____
\nAnswer:
\nWe know that,
\nA diagram that shows the prime factors of a composite number is a “factor tree”
\nHence, from the above,
\nWe can conclude that
\nThe term that is suitable for the given definition is: Factor tree<\/p>\n

Draw a line from each pair of numbers in Column A to the least common multiple (LCM) of the numbers in Column B<\/strong>.
\n\"Envision
\nAnswer:
\nThe given pair of numbers are: (9, 6), (9, 12), and (8, 7)
\nSo,
\nThe LCM of (9, 6) is: 18
\nThe LCM of (9, 12) is: 36
\nThe LCM of (8, 7) is: 56
\nHence, from the above,
\nWe can conclude that
\nThe matched pairs of Column A and Column B are:
\n\"\"<\/p>\n

Question 7.
\nLook at the variables in each expression below. Write Y if the terms of each expression are like terms. Write N if they are NOT like terms.<\/strong>
\na. 3a + 3z
\nb. \\(\\frac{x}{3}\\) + \\(\\frac{x}{4}\\)
\nc. 4j – j + 3.8j
\nAnswer:
\nThe given expressions are:
\na. 3a + 3z
\nb. \\(\\frac{x}{3}\\) + \\(\\frac{x}{4}\\)
\nc. 4j – j + 3.8j
\nNow,
\nWe know that,
\n“Like terms” are\u00a0terms whose variables (and their exponents such as the 2 in x2<\/sup>) are the same. In other words, terms that are “like” each other
\nHence, from the above,
\nWe can conclude that
\nThe representation of Y and N for the given expressions are:
\na. 3a + 3z\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0–\u00a0 N
\nb. \\(\\frac{x}{3}\\) + \\(\\frac{x}{4}\\)\u00a0 \u00a0 \u00a0 \u00a0 –\u00a0 Y
\nc. 4j – j + 3.8j\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 –\u00a0 Y<\/p>\n

Use Vocabulary in Writing<\/strong>
\nExplain one way to simplify the expression 4(3q – q). Use vocabulary words in your explanation.
\nAnswer:
\nThe given algebraic expression is: 4 (3q – q)
\nNow,
\nWhen we observe the given expression,
\nIn 3q, “3” is the coefficient
\n3q and q are the like terms
\nNow,
\n4 (3q – q) = 4 (3q – 1q)
\n= 4q (3 – 1)
\n= 4q (2)
\n= 8q
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: 8q<\/p>\n

Concepts and Skills Review<\/strong><\/p>\n

LESSON 3.1 Understand and Represent exponents<\/h3>\n

Quick Review<\/strong>
\nAn exponent is a way to show repeated multiplication.<\/p>\n

Example<\/strong>
\nUse an exponent to write the expression 6 \u00d7 6 \u00d7 6. Then evaluate the expression.
\n6 is used as a factor 3 times.
\n6 is the base and 3 is the exponent.
\n6 \u00d7 6 \u00d7 6 = 63<\/sup> = 216<\/p>\n

Find 60<\/sup><\/p>\n

A number with an exponent of 0 is always equal to 1.
\n60<\/sup> = 1<\/p>\n

Practice<\/strong><\/p>\n

Write each expression using an exponent.<\/strong><\/p>\n

Question 1.
\n8 \u00d7 8 \u00d7 8 \u00d7 8 \u00d7 8 \u00d7 8 \u00d7 8
\nAnswer:
\nThe given repeated multiplication is: 8 \u00d7 8 \u00d7 8 \u00d7 8 \u00d7 8 \u00d7 8 \u00d7 8
\nNow,
\nWe know that,
\nAn “Exponent” tells the number of times the base i.e., the given number is used as afactor
\nSo,
\n8 \u00d7 8 \u00d7 8 \u00d7 8 \u00d7 8 \u00d7 8 \u00d7 8 = 87<\/sup>
\nHence, from the above,
\nWe can conclude that
\nThe representation of the given expression using an exponent is: 87<\/sup><\/p>\n

Question 2.
\n4
\nAnswer:
\nThe given repeated multiplication is: 4
\nNow,
\nWe know that,
\nAn “Exponent” tells the number of times the base i.e., the given number is used as afactor
\nSo,
\n4 = 41<\/sup>
\nHence, from the above,
\nWe can conclude that
\nThe representation of the given expression using an exponent is: 41<\/sup><\/p>\n

Question 3.
\n10 \u00d7 10 \u00d7 10 \u00d7 10
\nAnswer:
\nThe given repeated multiplication is: 10 \u00d7 10 \u00d7 10 \u00d7 10
\nNow,
\nWe know that,
\nAn “Exponent” tells the number of times the base i.e., the given number is used as afactor
\nSo,
\n10 \u00d7 10 \u00d7 10 \u00d7 10 = 104<\/sup>
\nHence, from the above,
\nWe can conclude that
\nThe representation of the given expression using an exponent is: 104<\/sup><\/p>\n

Evaluate each expression.<\/strong><\/p>\n

Question 4.
\n92<\/sup>
\nAnswer:
\nThe given expression is: 9\u00b2
\nNow,
\nWe know that,
\nAn “Exponent” tells the number of times the base is used as a factor
\nSo,
\n9\u00b2 = 9 \u00d7 9
\n= 81
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: 81<\/p>\n

Question 5.
\n991<\/sup>
\nAnswer:
\nThe given expression is: 991<\/sup>
\nNow,
\nWe know that,
\nAn “Exponent” tells the number of times the base is used as a factor
\nSo,
\n991<\/sup> = 99
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: 99<\/p>\n

Question 6.
\n3,1050<\/sup>
\nAnswer:
\nThe given expression is: 3,1050<\/sup>
\nNow,
\nWe know that,
\nAn “Exponent” tells the number of times the base is used as a factor
\nAny nonzero number raised to an exponent of zero has always a value of 1
\nSo,
\n3,1050<\/sup> = 1
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: 1<\/p>\n

Question 7.
\n222<\/sup>
\nAnswer:
\nThe given expression is: 222<\/sup>
\nNow,
\nWe know that,
\nAn “Exponent” tells the number of times the base is used as a factor
\nSo,
\n222<\/sup> = 22 \u00d7 22
\n= 484
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: 484<\/p>\n

Question 8.
\n27<\/sup>
\nAnswer:
\nThe given expression is: 27<\/sup>
\nNow,
\nWe know that,
\nAn “Exponent” tells the number of times the base is used as a factor
\nSo,
\n27<\/sup> = 2 \u00d7 2 \u00d7 2 \u00d7 2 \u00d7 2 \u00d7 2 \u00d7 2
\n= 128
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: 128<\/p>\n

Question 9.
\n34<\/sup>
\nAnswer:
\nThe given expression is: 34<\/sup>
\nNow,
\nWe know that,
\nAn “Exponent” tells the number of times the base is used as a factor
\nSo,
\n34<\/sup> = 3 \u00d7 3 \u00d7 3 \u00d7 3
\n= 81
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: 81<\/p>\n

LESSON 3.2 Find Greatest Common Factor and Least Common Multiple<\/strong><\/p>\n

Quick Review<\/strong>
\nYou can use prime factorization to find the greatest common factor and the least common multiple of two numbers.<\/p>\n

Example<\/strong>
\nFind the greatest common factor (GCF) and the least common multiple (LCM) of 12 and 6.<\/p>\n

List the prime factors of both numbers.
\n\"Envision<\/p>\n

Practice<\/strong>
\nFind the GCF for each pair of numbers. Use the GCF and the Distributive Property to find the sum of each pair of numbers.<\/strong><\/p>\n

Question 1.
\n30, 100
\nAnswer:
\nThe given numbers are: 30, 100
\nNow,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe GCF of the given numbers is: 10<\/p>\n

Question 2.
\n8, 52
\nAnswer:
\nThe given numbers are: 8, 52
\nNow,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe GCF of the given numbers is: 4<\/p>\n

Question 3.
\n28, 42
\nAnswer:
\nThe given numbers are: 28, 42
\nNow,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe GCF of the given numbers is: 14<\/p>\n

Question 4.
\n37, 67
\nAnswer:
\nThe given numbers are: 37, 67
\nNow,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe GCF of the given numbers is: 1<\/p>\n

Question 5.
\n12, 24
\nAnswer:
\nThe given numbers are: 12, 24
\nNow,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe GCF of the given numbers is: 12<\/p>\n

Question 6.
\n8, 12
\nAnswer:
\nThe given numbers are: 8, 12
\nNow,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe GCF of the given numbers is: 4<\/p>\n

Find the LCM for each pair of numbers.<\/strong><\/p>\n

Question 7.
\n4, 9
\nAnswer:
\nThe given numbers are: 4, 9
\nNow,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe LCM of the given numbers is: 36<\/p>\n

Question 8.
\n3, 6
\nAnswer:
\nThe given numbers are: 3, 6
\nNow,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe LCM of the given numbers is: 6<\/p>\n

Question 9.
\n8, 10
\nAnswer:
\nThe given numbers are: 8, 10
\nNow,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe LCM of the given numbers is: 40<\/p>\n

Question 10.
\n3, 5
\nAnswer:
\nThe given numbers are: 3, 5
\nNow,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe LCM of the given numbers is: 15<\/p>\n

Question 11.
\n12, 5
\nAnswer:
\nThe given numbers are: 12, 5
\nNow,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe LCM of the given numbers is: 60<\/p>\n

Question 12.
\n4, 11
\nAnswer:
\nThe given numbers are: 4, 11
\nNow,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe LCM of the given numbers is: 44<\/p>\n

LESSON 3.3 Write and Evaluate Numerical expressions<\/strong><\/p>\n

Quick Review<\/strong>
\nUse the order of operations to evaluate numerical expressions.<\/p>\n

Example<\/strong>
\nEvaluate the expression 32<\/sup> + 2[(21 \u2013 9) \u00f7 4].
\n\"Envision<\/p>\n

Practice<\/strong>
\nEvaluate each expression.<\/strong><\/p>\n

Question 1.
\n80 \u2013 42<\/sup> \u00f7 8
\nAnswer:
\nThe given expression is: 80 \u2013 42<\/sup> \u00f7 8
\nNow,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: 78<\/p>\n

Question 2.
\n92.3 \u2013 (3.2 \u00f7 0.4) \u00d7 23<\/sup>
\nAnswer:
\nThe given expression is: 92.3 \u2013 (3.2 \u00f7 0.4) \u00d7 23<\/sup>
\nNow,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: 28.3<\/p>\n

Question 3.
\n[(23<\/sup> \u00d7 2.5) \u00f7 \\(\\frac{1}{2}\\)] + 120
\nAnswer:
\nThe given expression is: [(23<\/sup> \u00d7 2.5) \u00f7 \\(\\frac{1}{2}\\)] + 120
\nNow,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: 160<\/p>\n

Question 4.
\n[20 + (2.5 \u00d7 3)] – 33<\/sup>
\nAnswer:
\nThe given expression is: [20 + (2.5 \u00d7 3)] – 33<\/sup>
\nNow,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: 0.5<\/p>\n

Question 5.
\n\"Envision
\nAnswer:
\nThe given expression is: [(2 \u00d7 100<\/sup> ) \u00f7 \\(\\frac{1}{3}\\)] + 8
\nNow,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: 14<\/p>\n

LESSONS 3.4 AND 3.5 Write and Evaluate Algebraic expressions<\/strong><\/p>\n

Quick Review<\/strong>
\nAn algebraic expression can be written to represent a situation with an unknown quantity. Use a variable to represent the unknown quantity. An algebraic expression can be evaluated by substituting a value for the variable and performing the operations.<\/p>\n

Example<\/strong>
\nWrite an algebraic expression for 9 times the difference of 12 and a divided by 2. Then evaluate the expression for a = 4.
\n9 times the difference of 12 and a divided by 2″ is represented by 9 \u00d7 (12 – a) \u00f7 2.
\n\"Envision<\/p>\n

Practice<\/strong>
\nWrite an algebraic expression to represent each situation.<\/strong><\/p>\n

Question 1.
\n22 less than 5 times a number f
\nAnswer:
\nThe given situation is: 22 less than 5 times a number f
\nNow,
\nWe know that,
\nThe term “Less than” represents the subtraction
\nThe term”many times” (or) “times” represents the multiplication
\nHence, from the above,
\nWe can conclude that
\nThe representation of the given situation in the form of an algebraic expression is: 5f – 22<\/p>\n

Question 2.
\n48 times a number of game markers, g
\nAnswer:
\nThe given situation is: 48 times a number of game markers, g
\nNow,
\nWe know that,
\nThe term”many times” (or) “times” represents the multiplication
\nHence, from the above,
\nWe can conclude that
\nThe representation of the given situation in the form of an algebraic expression is: 48g<\/p>\n

Question 3.
\nA number of eggs, e, divided by 12
\nAnswer:
\nThe given situation is: A number of eggs, e, divided by 12
\nNow,
\nWe know that,
\nThe term”Division” (or) “Divide” represents the division
\nHence, from the above,
\nWe can conclude that
\nThe representation of the given situation in the form of an algebraic expression is: \\(\\frac{e}{12}\\)<\/p>\n

Question 4.
\n3 times the sum of m and 7
\nAnswer:
\nThe given situation is: 3 times the sum of m and 7
\nNow,
\nWe know that,
\nThe term”many times” (or) “times” represents the multiplication
\nThe term “Sum” (or) “Add” represents the addition
\nHence, from the above,
\nWe can conclude that
\nThe representation of the given situation in the form of an algebraic expression is: 3 (m + 7)<\/p>\n

Evaluate each expression for n = 7, x = 4, y = 8, and z = 1.<\/strong><\/p>\n

Question 5.
\n12x – 7
\nAnswer:
\nThe given expression is: 12x – 7
\nNow,
\nFor x = 4,
\n12x – 7 = 12 (4) – 7
\n= 48 – 7
\n= 41
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: 41<\/p>\n

Question 6.
\nx2<\/sup> \u00f7 y
\nAnswer:
\nThe given expression is: x\u00b2 \u00f7 y
\nNow,
\nFor x = 4, and y = 8,
\nx\u00b2 \u00f7 y = 4\u00b2 \u00f7 8
\n= 16 \u00f7 8
\n= 2
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: 2<\/p>\n

Question 7.
\n5z + 3n – z3<\/sup>
\nAnswer:
\nThe given expression is: 5z + 3n – z3<\/sup>
\nNow,
\nFor z = 1, and n = 7,
\n5z + 3n – z3<\/sup> = 5 (1) + 3 (7) – 13<\/sup>
\n= 5 + 21 – 1
\n= 25
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: 25<\/p>\n

Question 8.
\ny2<\/sup> \u00f7 2x + 3n – z
\nAnswer:
\nThe given expression is: y2<\/sup> \u00f7 2x + 3n – z
\nNow,
\nFor x = 4, n = 7, z = 1, and y = 8,
\ny2<\/sup> \u00f7 2x + 3n – z = 8\u00b2 \u00f7 2 (4) + 3 (7) – 1
\n= 64 \u00f7 8 + 21 – 1
\n= 8 + 20
\n= 28
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: 28<\/p>\n

LESSON 3.6 Generate Equivalent expressions<\/strong><\/p>\n

Quick Review<\/strong>
\nEquivalent expressions are expressions that have the same value. The properties of operations and substitution can be used to write and identify equivalent expressions.<\/p>\n

Example<\/strong>
\nAre the expressions 5x + 20, 5(x + 4), and x + 4 equivalent?
\nFor algebraic expressions to be equivalent, each expression must name the same value no matter what value is substituted for the variable.
\n\"Envision
\nUse the Distributive Property to write
\n5x + 20 as 5(x + 4).
\n5x + 20 = 5 \u2022 x + 5 \u2022 4
\n= 5(x + 4)
\nProperties of operations cannot be used to write 5x + 20 or 5(x + 4) as x + 4.
\n5x + 20 and 5(x + 4) are equivalent expressions.<\/p>\n

Practice<\/strong>
\nComplete the table. Then circle the expressions that are equivalent.<\/strong><\/p>\n

Question 1.
\n\"Envision
\nAnswer:
\nThe given expressions are:
\na. 5 (2.2y + 1) – 3\u00a0 \u00a0b. 11y + 5 – y\u00a0 \u00a0c. 11y + 2
\nNow,
\na.
\nFor y = 1,
\n5 (2.2y + 1) – 3 = 5 (2.2 + 1) – 3
\n= 13
\nFor y = 2,
\n5 (2.2y + 1) = 5 (2.2 (2) + 1) – 3
\n= 24
\nFor y = 3,
\n5 (2.2y + 1) – 3 = 5 (2.2 (3) + 1) – 3
\n= 35
\nNow,
\nb.
\nFor y = 1,
\n11y + 5 – y = 11 + 5 – 1
\n= 15
\nFor y = 2,
\n11y + 5 – y = 22 + 5 – 2
\n= 25
\nFor y = 3,
\n11y + 5 – y = 33 + 5 – 3
\n= 35
\nNow,
\nc.
\nFor y = 1,
\n11y + 2 = 11 + 2
\n= 13
\nFor y = 2,
\n11y + 2 = 22 + 2
\n= 24
\nFor y = 3,
\n11y + 2 = 33 + 2
\n= 35
\nHence, from the above,
\nWe can conclude that
\nThe completed table and the expressions (Circled in the table) that are equivalent is:
\n\"\"<\/p>\n

In 2-4, write Yes or No to indicate whether the expressions are equivalent.<\/strong><\/p>\n

Question 2.
\n10x \u2013 3 + 2x – 5 and 4(3x \u2013 2)
\nAnswer:
\nThe given expressions are: 10x \u2013 3 + 2x – 5 and 4(3x \u2013 2)
\nNow,
\n10x – 3 + 2x – 5 = (10x + 2x) – (3 + 5)
\n= 12x – 8
\n= 4 (3x – 2)
\nSo,
\n10x – 3 + 2x – 5 = 4 (3x – 2)
\nHence, from the above,
\nWe can conclude that
\nWe can indicate YES since the given expressions are equivalent<\/p>\n

Question 3.
\n3y + 3 and 9(y + \\(\\frac{1}{3}\\))
\nAnswer:
\nThe given expressions are: 3y + 3 and 9(y + \\(\\frac{1}{3}\\))
\nNow,
\n9(y + \\(\\frac{1}{3}\\)) = 9 (y) + 9 (\\(\\frac{1}{3}\\))
\n= 9y + 3
\nSo,
\n3y + 3 \u2260 9y + 3
\nHence, from the above,
\nWe can conclude that
\nWe can indicate NO since the given expressions are not equivalent<\/p>\n

Question 4.
\n6(3x + 1) and 9x + 6 + 9x
\nAnswer:
\nThe given expressions are: 6(3x + 1) and 9x + 6 + 9x
\nNow,
\n9x + 6 + 9x = (9x + 9x) + 6
\n= 18x + 6
\n= 6 (3x + 1)
\nSo,
\n6 (3x + 1) = 9x + 6 + 9x
\nHence, from the above,
\nWe can conclude that
\nWe can indicate YES since the given expressions are equivalent<\/p>\n

In 5-7, use properties of operations to complete the equivalent expressions.<\/strong><\/p>\n

Question 5.
\n2(x + 4) and ___x + ____
\nAnswer:
\nThe given expression is: 2 (x + 4)
\nNow,
\nBy using the Distributive Property,
\n2 (x + 4) = 2 (x) + 2 (4)
\n= 2x + 8
\nHence, from the above,
\nWe can conclude that
\nThe equivalent expression for the given expression is: 2x + 8<\/p>\n

Question 6.
\n5x \u2013 45 and 5(___ – ___ )
\nAnswer:
\nThe given expression is: 5x – 45
\nNow,
\nBy using the Distributive Property,
\n5x – 45 = 5 (x – 9)
\nHence, from the above,
\nWe can conclude that
\nThe equivalent expression for the given expression is: 5 (x – 9)<\/p>\n

Question 7.
\n3(x + 7) and ___ \u00d7+ ____
\nAnswer:
\nThe given expression is: 3 (x + 7)
\nNow,
\nBy using the Distributive Property,
\n3 (x + 7) = 3 (x) + 3 (7)
\n= 3x + 21
\nHence, from the above,
\nWe can conclude that
\nThe equivalent expression for the given expression is: 3x + 21<\/p>\n

LESSON 3.7 Simplify Algebraic expressions<\/strong><\/p>\n

Quick Review<\/strong>
\nCombine like terms to simplify algebraic expressions.<\/p>\n

Example<\/strong>
\nSimplify the expression 3x + 7 + 6x.
\n\"Envision<\/p>\n

Practice<\/strong>
\nSimplify each expression.<\/strong><\/p>\n

Question 1.
\n9y + 4 – 6y
\nAnswer:
\nThe given expression is: 9y + 4 – 6y
\nNow,
\n9y + 4 – 6y
\n= (9y – 6y) + 4
\n= 5y + 4
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: 5y + 4<\/p>\n

Question 2.
\n3x + 5 + 7x
\nAnswer:
\nThe given expression is: 3x + 5 + 7x
\nNow,
\n3x + 5 + 7x
\n= (3x + 7x) + 5
\n= 10x + 5
\n= 5 (2x + 1)
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: 5 (2x + 1)<\/p>\n

Question 3.
\n8x + 13 \u2013 3x + 9
\nAnswer:
\nThe given expression is: 8x + 13 – 3x + 9
\nNow,
\n8x + 13 – 3x + 9
\n= (8x – 3x) + (13 + 9)
\n= 5x + 22
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: 5x + 22<\/p>\n

Question 4.
\ny2<\/sup> + 3y2<\/sup>
\nAnswer:
\nThe given expression is: y\u00b2 + 3y\u00b2
\nNow,
\ny\u00b2 + 3y\u00b2
\n= y\u00b2 (1 + 3)
\n= y\u00b2 (4)
\n= 4y\u00b2
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: 4y\u00b2<\/p>\n

Question 5.
\n4x + 15 – 3x + 10
\nAnswer:
\nThe given expression is: 4x + 15 – 3x + 10
\nNow,
\n4x + 15 – 3x + 10
\n= (4x – 3x) + (15 + 10)
\n= x + 25
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: x + 25<\/p>\n

Question 6.
\n10x + 2x \u2013 12x
\nAnswer:
\nThe given expression is: 10x + 2x – 12x
\nNow,
\n10x + 2x – 12x
\n= x (10 + 2 – 12)
\n= x (12 – 12)
\n= x (0)
\n= 0
\nHence, from the above,
\nWe can conclude that
\nThe value of the given expression is: 0<\/p>\n

Topic 3 Fluency Practice<\/h3>\n

Crisscrossed<\/strong><\/p>\n

\"Envision<\/p>\n","protected":false},"excerpt":{"rendered":"

Practice with the help of enVision Math Common Core Grade 6 Answer Key Topic 3 Numeric and Algebraic Expressions regularly and improve your accuracy in solving questions. Envision Math Common Core 6th Grade Answers Key Topic 3 Numeric And Algebraic Expressions ?Topic essential Question\u00a0 What are expressions and how they can be written and evaluated? … Read more<\/a><\/p>\n","protected":false},"author":13,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"spay_email":""},"categories":[10],"tags":[],"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/posts\/90153"}],"collection":[{"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/users\/13"}],"replies":[{"embeddable":true,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/comments?post=90153"}],"version-history":[{"count":1,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/posts\/90153\/revisions"}],"predecessor-version":[{"id":90154,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/posts\/90153\/revisions\/90154"}],"wp:attachment":[{"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/media?parent=90153"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/categories?post=90153"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/tags?post=90153"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}