Big Ideas Math

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Big Ideas Math Book Algebra 2 Answer Key Chapter 6 Exponential and Logarithmic Functions

BIM Algebra 2 Ch 6 Exponential and Logarithmic Functions covers Questions from Exercises 6.1 to 6.7, Review Tests, Chapter Tests, Cumulative Assessments, etc. With the help of this ultimate preparation guide provided performance task test, students can check their knowledge on Chapter 6 Exponential and Logarithmic Functions concepts. Tap on the link directly and practice each & every lesson in Ch 6 Big Ideas Math Algebra 2 Solution Key Pdf. Access online Big Ideas Math Book Algebra 2 Ch 6 Exponential and Logarithmic Functions Answers pdf without paying a single penny.

• Exponential and Logarithmic Functions Maintaining Mathematical Proficiency – Page 293
• Exponential and Logarithmic Functions Mathematical Practices – Page 294
• Lesson 6.1 Exponential Growth and Decay Functions – Page(295-302)
• Exponential Growth and Decay Functions 6.1 Exercises – Page(300-302)
• Lesson 6.2 The Natural Base e – Page(303-308)
• The Natural Base e 6.2 Exercises – Page(307-308)
• Lesson 6.3 Logarithms and Logarithmic Functions – Page(309-316)
• Logarithms and Logarithmic Functions 6.3 Exercises – Page(314-316)
• Lesson 6.4 Transformations of Exponential and Logarithmic Functions – Page(317-324)
• Transformations of Exponential and Logarithmic Functions 6.4 Exercises – Page(322-324)
• Exponential and Logarithmic Functions Study Skills: Forming a Weekly Study Group – Page 325
• Exponential and Logarithmic Functions 6.1–6.4 Quiz – Page 326
• Lesson 6.5 Properties of Logarithms – Page(327-332)
• Properties of Logarithms 6.5 Exercises -Page(331-332)
• Lesson 6.6 Solving Exponential and Logarithmic Equations -Page(333-340)
• Solving Exponential and Logarithmic Equations 6.6 Exercises – Page(338-340)
• Lesson 6.7 Modeling with Exponential and Logarithmic Functions – Page(341-348)
• Modeling with Exponential and Logarithmic Functions 6.7 Exercises – Page(346-348)
• Exponential and Logarithmic Functions Performance Task: Measuring Natural Disasters – Page 349
• Exponential and Logarithmic Functions Chapter Review – Page(350-352)
• Exponential and Logarithmic Functions Chapter Test – Page 353
• Exponential and Logarithmic Functions Cumulative Assessment – Page(354-355)

Exponential and Logarithmic Functions Maintaining Mathematical Proficiency

Evaluate the expression.
Question 1.
3 • 2⁴
Answer:

Question 2.
(−2)⁵
Answer:

Question 3.
−(\(\frac{5}{6}\))²
Answer:

Question 4.
(\(\frac{3}{4}\))³
Answer:

Find the domain and range of the function represented by the graph.
Question 5.

Answer:

Question 6.

Answer:

Question 7.

Answer:

Question 8.
ABSTRACT REASONING
Consider the expressions −4ⁿ and (−4)ⁿ, where n is an integer. For what values of n is each expression negative? positive? Explain your reasoning.
Answer:

Exponential and Logarithmic Functions Mathematical Practices

Mathematically proficient students know when it is appropriate to use general methods and shortcuts.

Monitoring Progress

Determine whether the data can be modeled by an exponential or linear function. Explain your reasoning. Then write the appropriate model and find y when x = 10
Question 1.

Answer:

Question 2.

Answer:

Question 3.

Answer:

Question 4.

Answer:

Lesson 6.1 Exponential Growth and Decay Functions

Essential Question What are some of the characteristics of the graph of an exponential function?You can use a graphing calculator to evaluate an exponential function. For example, consider the exponential function f(x) = 2^x

EXPLORATION 1

Identifying Graphs of Exponential Functions
Work with a partner. Match each exponential function with its graph. Use a table of values to sketch the graph of the function, if necessary.

EXPLORATION 2

Characteristics of Graphs of Exponential Functions
Work with a partner. Use the graphs in Exploration 1 to determine the domain, range, and y-intercept of the graph of f(x) = b^x, where b is a positive real number other than 1. Explain your reasoning.

Communicate Your Answer

Question 3.
What are some of the characteristics of the graph of an exponential function?
Answer:

Question 4.
In Exploration 2, is it possible for the graph of f (x) =b x to have an x-intercept? Explain your reasoning.
Answer:

Monitoring Progress

Tell whether the function represents exponential growth or exponential decay. Then graph the function.
Question 1.
y = 4^x
Answer:

Question 2.
y = (\(\frac{2}{3}\))^x
Answer:

Question 3.
f(x) = (0.25)^x
Answer:

Question 4.
f(x) = (1.5)^x
Answer:

Question 5.
WHAT IF?
In Example 2, the value of the car can be approximated by the model y = 25(0.9)^t. Identify the annual percent decrease in the value of the car. Estimate when the value of the car will be $8000.
Answer:

Question 6.
WHAT IF?
In Example 3, assume the world population increased by 1.5% each year. Write an equation to model this situation. Estimate the year when the world population was 7 billion.
Answer:

Question 7.
The amount y (in grams) of the radioactive isotope iodine-123 remaining after t hours is y = a(0.5)^t/13, where a is the initial amount (in grams). What percent of the iodine-123 decays each hour?
Answer:

Question 8.
WHAT IF?
In Example 5, find the balance after 3 years when the interest is compounded daily.
Answer:

Exponential Growth and Decay Functions 6.1 Exercises

Vocabulary and Core Concept Check
Question 1.
VOCABULARY
In the exponential growth model y = 2.4(1.5)^x, identify the initial amount, the growth factor, and the percent increase.
Answer:

Question 2.
WHICH ONE DOESN’T BELONG?
Which characteristic of an exponential decay function does not belong with the other three? Explain your reasoning

Answer:

Monitoring Progress and Modeling with Mathematics

In Exercises 3–8, evaluate the expression for (a) x = -2 and (b) x = 3.
Question 3.
2^x
Answer:

Question 4.
4^x
Answer:

Question 5.
8 • 3^x
Answer:

Question 6.
6 • 2^x
Answer:

Question 7.
5 + 3^x
Answer:

Question 8.
2^x − 2
Answer:

In Exercises 9–18, tell whether the function represents exponential growth or exponential decay. Then graph the function.
Question 9.
y = 6^x
Answer:

Question 10.
y = 7^x
Answer:

Question 11.
y =(\(\frac{1}{6}\))^x
Answer:

Question 12.
y =(\(\frac{1}{8}\))^x
Answer:

Question 13.
y = (\(\frac{4}{3}\))^x
Answer:

Question 14.
y = (\(\frac{2}{5}\))^x
Answer:

Question 15.
y = (1.2)^x
Answer:

Question 16.
y = (0.75)^x
Answer:

Question 17.
y = (0.6)^x
Answer:

Question 18.
y = (1.8)^x
Answer:

ANALYZING RELATIONSHIPS In Exercises 19 and 20, use the graph of f(x) = b^x to identify the value of the base b.
Question 19.

Answer:

Question 20.

Answer:

Question 21.
MODELING WITH MATHEMATICS
The value of a mountain bike y (in dollars) can be approximated by the model y = 200(0.75)^t, where t is the number of years since the bike was new.
a. Tell whether the model represents exponential growth or exponential decay.
b. Identify the annual percent increase or decrease in the value of the bike.
c. Estimate when the value of the bike will be $50.
Answer:

Question 22.
MODELING WITH MATHEMATICS
The population P(in thousands) of Austin, Texas, during a recent decade can be approximated by y = 494.29(1.03)^t, where t is the number of years since the beginning of the decade.
a. Tell whether the model represents exponential growth or exponential decay.
b. Identify the annual percent increase or decrease in population.
c. Estimate when the population was about 590,000.
Answer:

Question 23.
MODELING WITH MATHEMATICS
In 2006, there were approximately 233 million cell phone subscribers in the United States. During the next 4 years, the number of cell phone subscribers increased by about 6% each year.
a. Write an exponential growth model giving the number of cell phone subscribers y (in millions) t years after 2006. Estimate the number of cell phone subscribers in 2008.
b. Estimate the year when the number of cell phone subscribers was about 278 million.
Answer:

Question 24.
MODELING WITH MATHEMATICS
You take a 325 milligram dosage of ibuprofen. During each subsequent hour, the amount of medication in your bloodstream decreases by about 29% each hour.
a. Write an exponential decay model giving the amount y (in milligrams) of ibuprofen in your bloodstream t hours after the initial dose.
b. Estimate how long it takes for you to have 100milligrams of ibuprofen in your bloodstream.
Answer:

JUSTIFYING STEPS In Exercises 25 and 26, justify each step in rewriting the exponential function.
Question 25.

Answer:

Question 26.

Answer:

Question 27.
PROBLEM SOLVING
When a plant or animal dies, it stops acquiring carbon-14 from the atmosphere. The amount y (in grams) of carbon-14 in the body of an organism after t years is y = a(0.5)^t/5730, where a is the initial amount (in grams). What percent of the carbon-14 is released each year?
Answer:

Question 28.
PROBLEM SOLVING
The number y of duckweed fronds in a pond after t days is y = a(1230.25)^t/16, where a is the initial number of fronds. By what percent does the duckweed increase each day?

Answer:

In Exercises 29–36, rewrite the function in the form y = a(1 + r)^t or y = a(1 – r)^t. Then state the growth or decay rate.
Question 29.
y = a(2)^t/3
Answer:

Question 30.
y = a(4)^t/6
Answer:

Question 31.
y = a(0.5)^t/12
Answer:

Question 32.
y = a(0.25)^t/9
Answer:

Question 33.
y = a(\(\frac{2}{3}\))^t/10
Answer:

Question 34.
y = a(\(\frac{5}{4}\))^t/22
Answer:

Question 35.
y = a(2)^8t
Answer:

Question 36.
y = a(\(\frac{1}{3}\))^3t
Answer:

Question 37.
PROBLEM SOLVING
You deposit $5000 in an account that pays 2.25% annual interest. Find the balance after 5 years when the interest is compounded quarterly.
Answer:

Question 38.
DRAWING CONCLUSIONS
You deposit $2200 into three separate bank accounts that each pay 3% annual interest. How much interest does each account earn after 6 years?

Answer:

Question 39.
ERROR ANALYSIS
You invest $500 in the stock of a company. The value of the stock decreases 2% each year. Describe and correct the error in writing a model for the value of the stock after t years.

Answer:

Question 40.
ERROR ANALYSIS
You deposit $250 in an account that pays 1.25% annual interest. Describe and correct the error in finding the balance after 3 years when the interest is compounded quarterly.

Answer:

In Exercises 41– 44, use the given information to find the amount A in the account earning compound interest after 6 years when the principal is $3500.
Question 41.
r = 2.16%, compounded quarterly
Answer:

Question 42.
r = 2.29%, compounded monthly
Answer:

Question 43.
r = 1.83%, compounded daily
Answer:

Question 44.
r = 1.26%, compounded monthly
Answer:

Question 45.
USING STRUCTURE
A website recorded the number y of referrals it received from social media websites over a 10-year period. The results can be modeled by y = 2500(1.50)^t, where t is the year and 0 ≤ t≤ 9. Interpret the values of a and b in this situation. What is the annual percent increase? Explain.
Answer:

Question 46.
HOW DO YOU SEE IT?
Consider the graph of an exponential function of the form f (x) = ab^x.

a. Determine whether the graph of f represents exponential growth or exponential decay.
b. What are the domain and range of the function? Explain.
Answer:

Question 47.
MAKING AN ARGUMENT
Your friend says the graph of f(x) = 2^x increases at a faster rate than the graph of g (x) = x² when x ≥ 0. Is your friend correct? Explain your reasoning.

Answer:

Question 48.
THOUGHT PROVOKING
The function f(x) = b^x represents an exponential decay function. Write a second exponential decay function in terms of b and x.
Answer:

Question 49.
PROBLEM SOLVING
The population p of a small town after x years can be modeled by the function p = 6850(1.03)^x. What is the average rate of change in the population over the first 6 years? Justify your answer.
Answer:

Question 50.
REASONING
Consider the exponential function f(x) = ab^x.
a. Show that \(\frac{f(x+1)}{f(x)}\) = b.
b. Use the equation in part (a) to explain why there is no exponential function of the form f(x) = ab^x whose graph passes through the points in the table below.

Answer:

Question 51.
PROBLEM SOLVING
The number E of eggs a Leghorn chicken produces per year can be modeled by the equation E = 179.2(0.89)^w/52, where w is the age (in weeks) of the chicken and w ≥ 22.

a. Identify the decay factor and the percent decrease.
b. Graph the model.
c. Estimate the egg production of a chicken that is 2.5 years old.d. Explain how you can rewrite the given equation so that time is measured in years rather than in weeks.
Answer:

Question 52.
CRITICAL THINKING
You buy a new stereo for $1300 and are able to sell it 4 years later for $275. Assume that the resale value of the stereo decays exponentially with time. Write an equation giving the resale value V(in dollars) of the stereo as a function of the time t (in years) since you bought it.
Answer:

Maintaining Mathematical Proficiency

Simplify the expression.
Question 53.
x⁹ • x²
Answer:

Question 54.
\(\frac{x^{4}}{x^{3}}\)
Answer:

Question 55.
4x • 6x
Answer:

Question 56.
\(\left(\frac{4 x^{8}}{2 x^{6}}\right)⁴\)
Answer:

Question 57.
\(\frac{x+3 x}{2}\)
Answer:

Question 58.
\(\frac{6x}{2}\) + 4x
Answer:

Question 59.
\(\frac{12x}{4x}\) + 5x
Answer:

Question 60.
(2x • 3x⁵)³
Answer:

Lesson 6.2 The Natural Base e

Essential Question What is the natural base e?
So far in your study of mathematics, you have worked with special numbers such as π and i. Another special number is called the natural base and is denoted by e. The natural base e is irrational, so you cannot find its exact value.

EXPLORATION 1

Approximating the Natural Base e
Work with a partner. One way to approximate the natural base e is to approximate the sum
1 + \(\frac{1}{1}+\frac{1}{1 \cdot 2}+\frac{1}{1 \cdot 2 \cdot 3}+\frac{1}{1 \cdot 2 \cdot 3 \cdot 4}+\cdots\)
Use a spreadsheet or a graphing calculator to approximate this sum. Explain the steps you used. How many decimal places did you use in your approximation?

EXPLORATION 2

Approximating the Natural Base e
Work with a partner. Another way to approximate the natural base e is to consider the expression(1 + \(\frac{1}{x}\))^x .
As x increases, the value of this expression approaches the value of e. Copy and complete the table. Then use the results in the table to approximate e. Compare this approximation to the one you obtained in Exploration 1.

EXPLORATION 3

Graphing a Natural Base Function
Work with a partner. Use your approximate value of e in Exploration 1 or 2 to complete the table. Then sketch the graph of the natural base exponential function y =e^x. You can use a graphing calculator and the e^x key to check your graph. What are the domain and range of y = e^x? Justify your answers.

Communicate Your Answer

Question 4.
What is the natural base e?
Answer:

Question 5.
Repeat Exploration 3 for the natural base exponential function y = e^-x. Then compare the graph of y = e^x to the graph of y = e^-x.
Answer:

Question 6.
The natural base e is used in a wide variety of real-life applications. Use the Internet or some other reference to research some of the real-life applications of e.
Answer:

Monitoring Progress

Simplify the expression.
Question 1.
e⁷ • e⁴
Answer:

Question 2.
\(\frac{24 e^{8}}{8 e^{5}}\)
Answer:

Question 3.
(10e^-3x)³
Answer:

Tell whether the function represents exponential growth or exponential decay. Then graph the function.
Question 4.
y = \(\frac{1}{2}\)e^x
Answer:

Question 5.
y = 4e^-x
Answer:

Question 6.
f(x) = 2e^2x
Answer:

Question 7.
You deposit $4250 in an account that earns 5% annual interest compounded continuously. Compare the balance after 10 years with the accounts in Example 3.
Answer:

The Natural Base e 6.2 Exercises

Vocabulary and Core Concept Check
Question 1.
VOCABULARY
What is the natural base e?
Answer:

Question 2.
WRITING
Tell whether the function f(x) = \(\frac{1}{3}\)e^4x represents exponential growth or exponential decay. Explain.
Answer:

Monitoring Progress and Modeling with Mathematics

In Exercises 3–12, simplify the expression.
Question 3.
e³ • e⁵
Answer:

Question 4.
e^-4 • e⁶
Answer:

Question 5.
\(\frac{11 e^{9}}{22 e^{10}}\)
Answer:

Question 6.
\(\frac{27 e^{7}}{3 e^{4}}\)
Answer:

Question 7.
(5e^7x)⁴
Answer:

Question 8.
(4e^-2x)³
Answer:

Question 9.
\(\sqrt{9 e^{6 x}}\)
Answer:

Question 10.
\(\sqrt[3]{8 e^{12 x}}\)
Answer:

Question 11.
e^x • e^-6x • e⁸
Answer:

Question 12.
e^x • e⁴ • e^x+3
Answer:

ERROR ANALYSIS In Exercises 13 and 14, describe and correct the error in simplifying the expression.
Question 13.

Answer:

Question 14.

Answer:

In Exercises 15–22, tell whether the function represents exponential growth or exponential decay. Then graph the function.
Question 15.
y = e^3x
Answer:

Question 16.
y = e^-2x
Answer:

Question 17.
y = 2e^-x
Answer:

Question 18.
y = 3e^2x
Answer:

Question 19.
y = 0.5e^x
Answer:

Question 20.
y = 0.25e^-3x
Answer:

Question 21.
y = 0.4e^-0.25x
Answer:

Question 22.
y = 0.6e^0.5x
Answer:

ANALYZING EQUATIONS In Exercises 23–26, match the function with its graph. Explain your reasoning.
Question 23.
y = e^2x
Answer:

Question 24.
y = e^-2x
Answer:

Question 25.
y = 4e^-0.5x
Answer:

Question 26.
y = 0.75e^x
Answer:

USING STRUCTURE In Exercises 27–30, use the properties of exponents to rewrite the function in the form y = a(1 + r)^t or y = a(1 – r)^t. Then find the percent rate of change.
Question 27.
y = e^-0.25t
Answer:

Question 28.
y = e^-0.75t
Answer:

Question 29.
y = 2e^0.4t
Answer:

Question 30.
y = 0.5e^0.8t
Answer:

USING TOOLS In Exercises 31–34, use a table of values or a graphing calculator to graph the function. Then identify the domain and range.
Question 31.
y = e^x-2
Answer:

Question 32.
y = e^x+1
Answer:

Question 33.
y = 2e^x + 1
Answer:

Question 34.
y = 3e^x − 5
Answer:

Question 35.
MODELING WITH MATHEMATICS
Investment accounts for a house and education earn annual interest compounded continuously. The balance H(in dollars) of the house fund after t years can be modeled by H = 3224e0.05t. The graph shows the balance in the education fund over time. Which account has the greater principal? Which account has a greater balance after 10 years?

Answer:

Question 36.
MODELING WITH MATHEMATICS
Tritium and sodium-22 decay over time. In a sample of tritium, the amount y (in milligrams) remaining after t years is given by y = 10e^−0.0562t. The graph shows the amount of sodium-22 in a sample over time. Which sample started with a greater amount? Which has a greater amount after 10 years?

Answer:

Question 37.
OPEN-ENDED
Find values of a, b, r, and q such that f(x) = ae^rx and g(x) = be^qx are exponential decay functions, but \(\frac{f(x)}{g(x)}\) represents exponential growth.
Answer:

Question 38.
THOUGHT PROVOKING
Explain why A = P(1 + \(\frac{r}{n}\))^nt approximates A =Pe^rt as n approaches positive infinity.
Answer:

Question 39.
WRITING
Can the natural base e be written as a ratio of two integers? Explain.
Answer:

Question 40.
MAKING AN ARGUMENT
Your friend evaluates f(x) = e^-x when x = 1000 and concludes that the graph of y = fx) has an x-intercept at (1000, 0). Is your friend correct? Explain your reasoning.
Answer:

Question 41.
DRAWING CONCLUSIONS
You invest $2500 in an account to save for college. Account 1 pays 6% annual interest compounded quarterly. Account 2 pays 4% annual interest compounded continuously. Which account should you choose to obtain the greater amount in 10 years? Justify your answer.
Answer:

Question 42.
HOW DO YOU SEE IT?
Use the graph to complete each statement.

a. f(x) approaches ____ as x approaches +∞.
b. f(x) approaches ____ as x approaches −∞.
Answer:

Question 43.
PROBLEM SOLVING
The growth of Mycobacterium tuberculosis bacteria can be modeled by the function N(t) = ae^0.166t, where N is the number of cells after t hours and a is the number of cells when t = 0.
a. At 1:00 P.M., there are 30 M. tuberculosis bacteria in a sample. Write a function that gives the number of bacteria after 1:00 P.M.
b. Use a graphing calculator to graph the function in part (a).
c. Describe how to find the number of cells in the sample at 3:45 P.M.
Answer:

Maintaining Mathematical Proficiency

Write the number in scientific notation.
Question 44.
0.006
Answer:

Question 45.
5000
Answer:

Question 46.
26,000,000
Answer:

Question 47.
0.000000047
Answer:

Find the inverse of the function. Then graph the function and its inverse.
Question 48.
y = 3x + 5
Answer:

Question 49.
y = x² − 1, x ≤ 0
Answer:

Question 50.
y = \(\sqrt{x+6}\)
Answer:

Question 51.
y = x³ − 2
Answer:

Lesson 6.3 Logarithms and Logarithmic Functions

Essential Question What are some of the characteristics of the graph of a logarithmic function?
Every exponential function of the form f(x) = b^x, where b is a positive real number other than 1, has an inverse function that you can denote by g(x) = log_bx. This inverse function is called a logarithmic function with base b.

EXPLORATION 1

Rewriting Exponential Equations
Work with a partner. Find the value of x in each exponential equation. Explain your reasoning. Then use the value of x to rewrite the exponential equation in its equivalent logarithmic form, x = log_by.
a. 2^x = 8
b. 3^x = 9
c. 4^x = 2
d. 5^x = 1
e. 5^x = \(\frac{1}{5}\)
f. 8^x = 4

EXPLORATION 2

Graphing Exponential and Logarithmic Functions
Work with a partner. Complete each table for the given exponential function. Use the results to complete the table for the given logarithmic function. Explain your reasoning. Then sketch the graphs of f and g in the same coordinate plane.

EXPLORATION 3

Characteristics of Graphs of LogarithmicFunctions

Work with a partner. Use the graphs you sketched in Exploration 2 to determine the domain, range, x-intercept, and asymptote of the graph of g(x) = log_bx, where b is a positive real number other than 1. Explain your reasoning.

Communicate Your Answer

Question 4.
What are some of the characteristics of the graph of a logarithmic function?
Answer:

Question 5.
How can you use the graph of an exponential function to obtain the graph of a logarithmic function?
Answer:

Monitoring Progress

Rewrite the equation in exponential form.
Question 1.
log₃ 81 = 4
Answer:

Question 2.
log₇ 7 = 1
Answer:

Question 3.
log₁₄ 1 = 0
Answer:

Question 4.
log_1/2 32 = −5
Answer:

Rewrite the equation in logarithmic form.
Question 5.
7² = 49
Answer:

Question 6.
50⁰ = 1
Answer:

Question 7.
4^-1 = \(\frac{1}{4}\)
Answer:

Question 8.
256^1/8 = 2
Answer:

Evaluate the logarithm. If necessary, use a calculator and round your answer to three decimal places.
Question 9.
log₂ 32
Answer:

Question 10.
log₂₇ 3
Answer:

Question 11.
log 12
Answer:

Question 12.
ln 0.75
Answer:

Simplify the expression.
Question 13.
8^{log₈ x}
Answer:

Question 14.
log₇ 7^−3x
Answer:

Question 15.
log₂ 64^x
Answer:

Question 16.
e^{ln 20}
Answer:

Question 17.
Find the inverse of y = 4^x.
Answer:

Question 18.
Find the inverse of y = ln(x− 5).
Answer:

Graph the function.
Question 19.
y = log₂ x
Answer:

Question 20.
f(x) = log₅ x
Answer:

Question 21.
y = log_1/2 x
Answer:

Logarithms and Logarithmic Functions 6.3 Exercises

Vocabulary and Core Concept Check
Question 1.
COMPLETE THE SENTENCE
A logarithm with base 10 is called a(n) ___________ logarithm.
Answer:

Question 2.
COMPLETE THE SENTENCE
The expression log₃ 9 is read as ______________.
Answer:

Question 3.
WRITING
Describe the relationship between y = 7^x and y = log₇ x.
Answer:

Question 4.
DIFFERENT WORDS, SAME QUESTION
Which is different? Find “both” answers.

Answer:

Monitoring Progress and Modeling with Mathematics

In Exercises 5–10, rewrite the equation in exponential form.
Question 5.
log₃ 9 = 2
Answer:

Question 6.
log₄ 4 = 1
Answer:

Question 7.
log₆ 1 = 0
Answer:

Question 8.
log₇ 343 = 3
Answer:

Question 9.
log_1/2 16 =−4
Answer:

Question 10.
log₃ \(\frac{1}{3}\) = −1
Answer:

In Exercises 11–16, rewrite the equation in logarithmic form.
Question 11.
6² = 36
Answer:

Question 12.
12⁰ = 1
Answer:

Question 13.
16^-1 = \(\frac{1}{16}\)
Answer:

Question 14.
5^-2 = \(\frac{1}{25}\)
Answer:

Question 15.
125^2/3 = 25
Answer:

Question 16.
49^1/2 = 7
Answer:

In Exercises 17–24, evaluate the logarithm.
Question 17.
log₃ 81
Answer:

Question 18.
log₇ 49
Answer:

Question 19.
log₃ 3
Answer:

Question 20.
log_1/2 1
Answer:

Question 21.
log₅ \(\frac{1}{625}\)
Answer:

Question 22.
log₈ \(\frac{1}{512}\)
Answer:

Question 23.
log₄ 0.25
Answer:

Question 24.
log₁₀ 0.001
Answer:

Question 25.
NUMBER SENSE
Order the logarithms from least value to greatest value.

Answer:

Question 26.
WRITING
Explain why the expressions log₂(−1) and log₁ 1 are not defined.
Answer:

In Exercises 27–32, evaluate the logarithm using a calculator. Round your answer to three decimal places.
Question 27.
log 6
Answer:

Question 28.
ln 12
Answer:

Question 29.
ln \(\frac{1}{3}\)
Answer:

Question 30.
log \(\frac{2}{7}\)
Answer:

Question 31.
3 ln 0.5
Answer:

Question 32.
log 0.6 + 1
Answer:

Question 33.
MODELING WITH MATHEMATICS
Skydivers use an instrument called an altimeter to track their altitude as they fall. The altimeter determines altitude by measuring air pressure. The altitude h (in meters) above sea level is related to the air pressure P(in pascals) by the function shown in the diagram. What is the altitude above sea level when the air pressure is 57,000 pascals?

Answer:

Question 34.
MODELING WITH MATHEMATICS
The pH value for a substance measures how acidic or alkaline the substance is. It is given by the formula pH = −log[H⁺], where H⁺ is the hydrogen ion concentration (in moles per liter). Find the pH of each substance.
a. baking soda: [H⁺] = 10^-8 moles per liter
b. vinegar: [H⁺] = 10^-3 moles per liter
Answer:

In Exercises 35–40, simplify the expression.
Question 35.
7^log₇x
Answer:

Question 36.
3^{log₃ 5x}
Answer:

Question 37.
e^ln4
Answer:

Question 38.
10^{log 15}
Answer:

Question 39.
log₃ 3^2x
Answer:

Question 40.
ln e^{x + 1}
Answer:

Question 41.
ERROR ANALYSIS
Describe and correct the error in rewriting 4⁻³ = \(\frac{1}{64}\) in logarithmic form.

Answer:

Question 42.
ERROR ANALYSIS
Describe and correct the error in simplifying the expression log₄ 64^x.

Answer:

In Exercises 43–52, find the inverse of the function.
Question 43.
y = 0.3^x
Answer:

Question 44.
y = 11^x
Answer:

Question 45.
y = log₂ x
Answer:

Question 46.
y = log_1/5 x
Answer:

Question 47.
y = ln(x − 1)
Answer:

Question 48.
y = ln 2x
Answer:

Question 49.
y = e^3x
Answer:

Question 50.
y = e^x-4
Answer:

Question 51.
y = 5^x − 9
Answer:

Question 52.
y = 13 + log x
Answer:

Question 53.
PROBLEM SOLVING
The wind speed s (in miles per hour) near the center of a tornado can be modeled by s = 93 log d + 65, where d is the distance (in miles) that the tornado travels.

a. In 1925, a tornado traveled 220 miles through three states. Estimate the wind speed near the center of the tornado.
b. Find the inverse of the given function. Describe what the inverse represents.
Answer:

Question 54.
MODELING WITH MATHEMATICS
The energy magnitude M of an earthquake can be modeled by M = \(\frac{2}{3}\)log E − 9.9, where E is the amount of energy released (in ergs).

a. In 2011, a powerful earthquake in Japan, caused by the slippage of two tectonic plates along a fault, released 2.24 × 10²⁸ ergs. What was the energy magnitude of the earthquake?
b. Find the inverse of the given function. Describe what the inverse represents.
Answer:

In Exercises 55–60, graph the function.
Question 55.
y = log₄ x
Answer:

Question 56.
y = log₆ x
Answer:

Question 57.
y = log_1/3 x
Answer:

Question 58.
y = log_1/4 x
Answer:

Question 59.
y = log₂ x − 1
Answer:

Question 60.
y = log₃(x + 2)
Answer:

USING TOOLS In Exercises 61–64, use a graphing calculator to graph the function. Determine the domain, range, and asymptote of the function.
Question 61.
y = log(x+ 2)
Answer:

Question 62.
y = −ln x
Answer:

Question 63.
y = ln(−x)
Answer:

Question 64.
y = 3 − log x
Answer:

Question 65.
MAKING AN ARGUMENT
Your friend states that every logarithmic function will pass through the point (1, 0). Is your friend correct? Explain your reasoning.
Answer:

Question 66.
ANALYZING RELATIONSHIPS
Rank the functions in order from the least average rate of change to the greatest average rate of change over the interval 1 ≤ x ≤ 10.
a. y = log₆ x
b. y = log_3/5 x

Answer:

Question 67.
PROBLEM SOLVING
Biologists have found that the length ℓ (in inches) of an alligator and its weight w (in pounds) are related by the function ℓ = 27.1 ln w − 32.8.

a. Use a graphing calculator to graph the function.
b. Use your graph to estimate the weight of an alligator that is 10 feet long.
c. Use the zero feature to find the x-intercept of the graph of the function. Does this x-value make sense in the context of the situation? Explain.
Answer:

Question 68.
HOW DO YOU SEE IT?
The figure shows the graphs of the two functions f and g.

a. Compare the end behavior of the logarithmic function g to that of the exponential function f.
b. Determine whether the functions are inverse functions. Explain.
c. What is the base of each function? Explain.
Answer:

Question 69.
PROBLEM SOLVING
A study in Florida found that the number s of fish species in a pool or lake can be modeled by the function s = 30.6 − 20.5 log A + 3.8(log A)²
where A is the area (in square meters) of the pool or lake.

a. Use a graphing calculator to graph the function on the domain 200 ≤ A ≤ 35,000.
b. Use your graph to estimate the number of species in a lake with an area of 30,000 square meters.
c. Use your graph to estimate the area of a lake that contains six species of fish.
d. Describe what happens to the number of fish species as the area of a pool or lake increases. Explain why your answer makes sense.
Answer:

Question 70.
THOUGHT PROVOKING
Write a logarithmic function that has an output of −4. Then sketch the graph of your function.
Answer:

Question 71.
CRITICAL THINKING
Evaluate each logarithm. (Hint: For each logarithm log_b x, rewrite b and x as powers of the same base.)
a. log₁₂₅ 25
b. log₈ 32
c. log₂₇ 81
d. log₄ 128
Answer:

Maintaining Mathematical Proficiency

Let f(x) = \(\sqrt [ 3 ]{ x }\). Write a rule for g that represents the indicated transformation of the graph of f.
Question 72.
g(x) = −f(x)
Answer:

Question 73.
g(x) = f (\(\frac{1}{2}\)x )
Answer:

Question 74.
g(x) = f(−x) + 3
Answer:

Question 75.
g(x) = f(x+ 2)
Answer:

Identify the function family to which f belongs. Compare the graph of f to the graph of its parent.
Question 76.

Answer:

Question 77.

Answer:

Question 78.

Answer:

Lesson 6.4 Transformations of Exponential and Logarithmic Functions

Essential Question How can you transform the graphs of exponential and logarithmic functions?

EXPLORATION 1

Identifying Transformations
Work with a partner. Each graph shown is a transformation of the parent function
f(x) = e^x or f(x) = ln x.
Match each function with its graph. Explain your reasoning. Then describe the transformation of f represented by g.
a. g(x) = e^x+2 − 3
b. g(x) = −e^x+2 + 1
c. g(x) = e^x-2 − 1
d. g(x) = ln(x + 2)
e. g(x) = 2 + ln x
f. g(x) = 2 + ln(−x)

EXPLORATION 2

Characteristics of Graphs
Work with a partner. Determine the domain, range, and asymptote of each function in Exploration 1. Justify your answers.

Communicate Your Answer

Question 3.
How can you transform the graphs of exponential and logarithmic functions?
Answer:

Question 4.
Find the inverse of each function in Exploration 1. Then check your answer by using a graphing calculator to graph each function and its inverse in the same viewing window.

Answer:

Monitoring Progress

Describe the transformation of f represented by g. Then graph each function.
Question 1.
f(x) = 2^x, g(x) = 2^x-3 + 1
Answer:

Question 2.
f(x) = e^-x, g(x) = e^-x − 5
Answer:

Question 3.
f(x) = 0.4^x, g(x) = 0.4^-2x
Answer:

Question 4.
f(x) = e^x, g(x) = −e^x+6
Answer:

Describe the transformation of f represented by g. Then graph each function.
Question 5.
f(x) = log₂x, g(x) = −3 log₂x
Answer:

Question 6.
f(x) = log_1/4x, g(x) = log_1/4(4x) − 5
Answer:

Question 7.
Let the graph of g be a horizontal stretch by a factor of 3, followed by a translation 2 units up of the graph of f(x) = e^-x. Write a rule for g.
Answer:

Question 8.
Let the graph of g be a reflection in the y-axis, followed by a translation 4 units to the left of the graph of f(x) = log x. Write a rule for g.
Answer:

Transformations of Exponential and Logarithmic Functions 6.4 Exercises

Vocabulary and Core Concept Check
Question 1.
WRITING
Given the function f(x) = ab^x-h + k, describe the effects of a, h, and k on the graph of the function.
Answer:

Question 2.
COMPLETE THE SENTENCE
The graph of g(x) = log4(−x) is a reflection in the __________ of the graph of f(x) = log₄x,
Answer:

Monitoring Progress and Modeling with Mathematics

In Exercises 3–6, match the function with its graph. Explain your reasoning.
Question 3.
f(x) = 2^x+2 − 2
Answer:

Question 4.
g(x) = 2^x+2 + 2
Answer:

Question 5.
h(x) = 2^x-2 − 2
Answer:

Question 6.
k(x) = 2^x-2 + 2
Answer:

In Exercises 7–16, describe the transformation of f represented by g. Then graph each function.
Question 7.
f(x) = 3^x, g(x) = 3^x + 5
Answer:

Question 8.
f(x) = 4^x, g(x) = 4^x − 8
Answer:

Question 9.
f(x) = e^x, g(x) = e^x − 1
Answer:

Question 10.
f(x) = e^x, g(x) = e^x + 4
Answer:

Question 11.
f(x) = 2^x, g(x) = 2^x-7
Answer:

Question 12.
f(x) = 5^x, g(x) = 5^x+1
Answer:

Question 13.
f(x) = e^-x, g(x) = e^-x + 6
Answer:

Question 14.
f(x) = e^-x, g(x) = e^-x − 9
Answer:

Question 15.
f(x) = (\(\frac{1}{4}\))^x, g (x) = (\(\frac{1}{4}\))^x-3 + 12
Answer:

Question 16.
f(x) = (\(\frac{1}{3}\))^x, g(x) = (\(\frac{1}{3}\))^x+2 − \(\frac{2}{3}\)
Answer:

In Exercises 17–24, describe the transformation of f represented by g. Then graph each function.
Question 17.
f(x) = e^x, g(x) = e^2x
Answer:

Question 18.
f(x) = e^x, g (x) =4—3 e^x
Answer:

Question 19.
f(x) = 2^x, g(x) = −2^x-3
Answer:

Question 20.
f(x) = 4^x, g(x) = 4^0.5x-5
Answer:

Question 21.
f(x) = e^-x, g(x) = 3e^-6x
Answer:

Question 22.
f(x) = e^-x, g(x) = e^-5x + 2
Answer:

Question 23.
f(x) = (\(\frac{1}{2}\))^x, g(x) = 6 (\(\frac{1}{2}\))^x+5 − 2
Answer:

Question 24.
f(x) = (\(\frac{3}{4}\))^x, g(x) = −(\(\frac{3}{4}\)) ^-7 + 1
Answer:

ERROR ANALYSIS In Exercises 25 and 26, describe and correct the error in graphing the function.
Question 25.
f(x) = 2^x + 3

Answer:

Question 26.
f(x) = 3^-x

Answer:

In Exercises 27–30, describe the transformation of f represented by g. Then graph each function.
Question 27.
f(x) = log₄ x, g(x) = 3 log₄ x − 5
Answer:

Question 28.
f(x) = log_1/3 x, g(x) = log_1/3(−x) + 6
Answer:

Question 29.
f(x) = log_1/5 x, g(x) = − log_1/5(x − 7)
Answer:

Question 30.
f(x) = log₂ x, g(x) = log₂(x + 2) − 3
Answer:

ANALYZING RELATIONSHIPS In Exercises 31–34, match the function with the correct transformation of the graph of f. Explain your reasoning.

Question 31.
y = f(x− 2)
Answer:

Question 32.
y = f(x + 2)
Answer:

Question 33.
y = 2f(x)
Answer:

Question 34.
y = f(2x)
Answer:

In Exercises 35–38, write a rule for g that represents the indicated transformations of the graph of f.
Question 35.
f(x) = 5^x; translation 2 units down, followed by a reflection in the y-axis
Answer:

Question 36.
f(x) = (\(\frac{2}{3}\))^x; reflection in the x-axis, followed by a vertical stretch by a factor of 6 and a translation 4 units left
Answer:

Question 37.
f(x) = e^x; horizontal shrink by a factor of \(\frac{1}{2}\), followed by a translation 5 units up
Answer:

Question 38.
f(x) =e^-x; translation 4 units right and 1 unit down, followed by a vertical shrink by a factor of \(\frac{1}{3}\)
Answer:

In Exercises 39–42, write a rule for g that represents the indicated transformation of the graph of f.
Question 39.
f(x) = log₆ x; vertical stretch by a factor of 6, followed by a translation 5 units down
Answer:

Question 40.
f(x) = log₂ x; reflection in the x-axis, followed by a translation 9 units left
Answer:

Question 41.
f(x) = log_1/2 x; translation 3 units left and 2 units up, followed by a reflection in the y-axis
Answer:

Question 42.
f(x) = ln x; translation 3 units right and 1 unit up, followed by a horizontal stretch by a factor of 8
Answer:

JUSTIFYING STEPS In Exercises 43 and 44, justify each step in writing a rule for g that represents the indicated transformations of the graph of f.
Question 43.
f(x) = log₇ x; reflection in the x-axis, followed by a translation 6 units down

Answer:

Question 44.
f(x) = 8x; vertical stretch by a factor of 4, followed by a translation 1 unit up and 3 units left

Answer:

USING STRUCTURE In Exercises 45–48, describe the transformation of the graph of f represented by the graph of g. Then give an equation of the asymptote.
Question 45.
f(x) = e^x, g(x) = e^x + 4
Answer:

Question 46.
f(x) = 3^x, g(x) = 3^x-9
Answer:

Question 47.
f(x) = ln x, g(x) = ln(x + 6)
Answer:

Question 48.
f(x) = log_1/5 x, g(x) = log_1/5 x + 13
Answer:

Question 49.
MODELING WITH MATHEMATICS
The slope Sof a beach is related to the average diameter d(in millimeters) of the sand particles on the beach by the equation S = 0.159 + 0.118 log d. Describe the transformation of f(d ) = log d represented by S. Then use the function to determine the slope of a beach for each sand type below.

Answer:

Question 50.
HOW DO YOU SEE IT?
The graphs of f(x) = b^x and g(x) = (\(\frac{1}{b}\))^x are shown for b = 2.

a. Use the graph to describe a transformation of the graph of f that results in the graph of g.
b. Does your answer in part (a) change when 0 < b < 1? Explain.
Answer:

Question 51.
MAKING AN ARGUMENT
Your friend claims a single transformation of f(x) = log x can result in a function g whose graph never intersects the graph of f. Is your friend correct? Explain your reasoning.
Answer:

Question 52.
THOUGHT PROVOKING
Is it possible to transform the graph of f(x) = e^x to obtain the graph of g(x) = ln x? Explain your reasoning.
Answer:

Question 53.
ABSTRACT REASONING
Determine whether each statement is always, sometimes, or never true. Explain your reasoning.
a. A vertical translation of the graph of f(x) = log x changes the equation of the asymptote.
b. A vertical translation of the graph of f(x) = e^x changes the equation of the asymptote.
c. A horizontal shrink of the graph of f(x) = log x does not change the domain.
d. The graph of g(x) = ab^x-h + k does not intersect the x-axis.
Answer:

Question 54.
PROBLEM SOLVING
The amount P (in grams) of 100 grams of plutonium-239 that remains after t years can be modeled by P= 100(0.99997)^t.
a. Describe the domain and range of the function.
b. How much plutonium-239 is present after 12,000 years?
c. Describe the transformation of the function if the initial amount of plutonium were 550 grams.
d. Does the transformation in part (c) affect the domain and range of the function? Explain your reasoning.
Answer:

Question 55.
CRITICAL THINKING
Consider the graph of the function h(x) = e^-x-2. Describe the transformation of the graph of f(x) = e^-x represented by the graph of h. Then describe the transformation of the graph of g(x) = e^x represented by the graph of h. Justify your answers.
Answer:

Question 56.
OPEN-ENDED
Write a function of the form y = ab^x-h + k whose graph has a y-intercept of 5 and an asymptote of y = 2.
Answer:

Maintaining Mathematical Proficiency

Perform the indicated operation.
Question 57.
Let f(x) = x⁴ and g(x) = x². Find ( fg)(x). Then evaluate the product when x = 3.
Answer:

Question 58.
Let f(x) = 4x⁶ and g(x) = 2x³. Find (\(\frac{f}{g}\)) (x). Then evaluate the quotient when x = 5.
Answer:

Question 59.
Let f(x) = 6x³ and g(x) = 8x³. Find ( f + g)(x). Then evaluate the sum when x = 2.
Answer:

Question 60.
Let f(x) = 2x^x and g(x) = 3x^x. Find ( f – g)(x). Then evaluate the difference when x = 6.
Answer:

Exponential and Logarithmic Functions Study Skills: Forming a Weekly Study Group

6.1–6.4 What Did You Learn?

Core Vocabulary

Core Concepts

Mathematical Practices
Question 1.
How did you check to make sure your answer was reasonable in Exercise 23 on page 300?
Answer:

Question 2.
How can you justify your conclusions in Exercises 23–26 on page 307?
Answer:

Question 3.
How did you monitor and evaluate your progress in Exercise 66 on page 315?
Answer:

Study Skills: Forming a Weekly Study Group

• Select students who are just as dedicated to doing well in the math class as you are.
• ind a regular meeting place that has minimal distractions.
• Compare schedules and plan at least one time a week to meet, allowing at least 1.5 hours for study time.
  

Exponential and Logarithmic Functions 6.1–6.4 Quiz

Tell whether the function represents exponential growth or exponential decay. Explain your reasoning.
Question 1.
f(x) = (4.25)^x
Answer:

Question 2.
y = (\(\frac{3}{8}\))^x
Answer:

Question 3.
y = e^0.6x
Answer:

Question 4.
f(x) = 5e^-2x
Answer:

Simplify the expression.
Question 5.
e⁸ • e⁴
Answer:

Question 6.
\(\frac{15 e^{3}}{3 e}\)
Answer:

Question 7.
(5e^4x)³
Answer:

Question 8.
el^{ln 9}
Answer:

Question 9.
log₇ 49^x
Answer:

Question 10.
log₃ 81^-2x
Answer:

Rewrite the expression in exponential or logarithmic form.
Question 11.
log₄ 1024 = 5
Answer:

Question 12.
log_1/3 27 = −3
Answer:

Question 13.
7⁴ = 2401
Answer:

Question 14.
4^-2 = 0.0625
Answer:

Evaluate the logarithm. If necessary, use a calculator and round your answer to three decimal places.
Question 15.
log 45
Answer:

Question 16.
ln 1.4
Answer:

Question 17.
log₂ 32
Answer:

Graph the function and its inverse.
Question 18.
f(x) = (\(\frac{1}{9}\))^x
Answer:

Question 19.
y = ln(x − 7)
Answer:

Question 20.
f(x) = log₅ (x+ 1)
Answer:

The graph of g is a transformation of the graph of f. Write a rule for g.
Question 21.
f(x) = log₃ x

Answer:

Question 22.
f(x) = 3^x

Answer:

Question 23.
f(x)= log_1/2 x

Answer:

Question 24.
You purchase an antique lamp for $150. The value of the lamp increases by 2.15% each year. Write an exponential model that gives the value y (in dollars) of the lamp t years after you purchased it.
Answer:

Question 25.
A local bank advertises two certificate of deposit (CD) accounts that you can use to save money and earn interest. The interest is compounded monthly for both accounts.

a. You deposit the minimum required amounts in each CD account. How much money is in each account at the end of its term? How much interest does each account earn? Justify your answers.
b. Describe the benefits and drawbacks of each account.
Answer:

Question 26.
The Richter scale is used for measuring the magnitude of an earthquake. The Richter magnitude R is given by R = 0.67 ln E + 1.17, where E is the energy (in kilowatt-hours) released by the earthquake. Graph the model. What is the Richter magnitude for an earthquake that releases 23,000 kilowatt-hours of energy?
Answer:

Lesson 6.5 Properties of Logarithms

Essential Question How can you use properties of exponents to derive properties of logarithms?
Let x = log_b m and y = log_b n.
The corresponding exponential forms of these two equations are
b^x = m and b^y = n.

EXPLORATION 1

Product Property of Logarithms
Work with a partner. To derive the Product Property, multiply m and n to obtain mn =b^xb^y = b^x+y.
The corresponding logarithmic form of mn = b^x+y is log_b mn = x + y. So,

EXPLORATION 2

Quotient Property of Logarithms
Work with a partner. To derive the Quotient Property, divide m by n to obtain
\(\frac{m}{n}=\frac{b^{x}}{b^{y}}\) = b^x-y.
The corresponding logarithmic form of \(\frac{m}{n}\) = b^x-y is log_b \(\frac{m}{n}\) = x − y. So,

EXPLORATION 3

Power Property of Logarithms
Work with a partner. To derive the Power Property, substitute b^x for m in the expression log_b mⁿ, as follows.

Communicate Your Answer

Question 4.
How can you use properties of exponents to derive properties of logarithms?
Answer:

Question 5.
Use the properties of logarithms that you derived in Explorations 1–3 to evaluate each logarithmic expression.
a. log₄ 16³
b. log₃ 81^-3
c. ln e² + ln e⁵
d. 2 ln e⁶ − ln e⁵
e. log₅ 75 − log₅ 3
f. log₄ 2 + log₄ 32
Answer:

Monitoring Progress

Use log₆ 5 ≈ 0.898 and log₆ 8 ≈ 1.161 to evaluate the logarithm.
Question 1.
log₆ \(\frac{5}{8}\)
Answer:

Question 2.
log₆ 40
Answer:

Question 3.
log₆ 64
Answer:

Question 4.
log₆ 125
Answer:

Expand the logarithmic expression.
Question 5.
log₆ 3x⁴
Answer:

Question 6.
ln \(\frac{5}{12x}\)
Answer:

Condense the logarithmic expression.
Question 7.
log x − log 9
Answer:

Question 8.
ln 4 + 3 ln 3 − ln 12
Answer:

Use the change-of-base formula to evaluate the logarithm.
Question 9.
log₅ 8
Answer:

Question 10.
log₈ 14
Answer:

Question 11.
log₂₆ 9
Answer:

Question 12.
log₁₂ 30
Answer:

Question 13.
WHAT IF?
In Example 6, the artist turns up the volume so that the intensity of the sound triples. By how many decibels does the loudness increase?
Answer:

Properties of Logarithms 6.5 Exercises

Vocabulary and Core Concept Check
Question 1.
COMPLETE THE SENTENCE
To condense the expression log₃ 2x + log₃ y, you need to use the __________ Property of Logarithms.
Answer:

Question 2.
WRITING
Describe two ways to evaluate log7 12 using a calculator.
Answer:

Monitoring Progress and Modeling with Mathematics

In Exercises 3–8, use log₇ 4 ≈ 0.712 and log₇ 12 ≈ 1.277 to evaluate the logarithm.
Question 3.
log₇ 3
Answer:

Question 4.
log₇ 48
Answer:

Question 5.
log₇ 16
Answer:

Question 6.
log₇ 64
Answer:

Question 7.
log₇ \(\frac{1}{4}\)
Answer:

Question 8.
log₇ \(\frac{1}{3}\)
Answer:

In Exercises 9–12, match the expression with the logarithm that has the same value. Justify your answer.

Answer:

In Exercises 13–20, expand the logarithmic expression.
Question 13.
log₃ 4x
Answer:

Question 14.
log₈ 3x
Answer:

Question 15.
log 10x⁵
Answer:

Question 16.
ln 3x⁴
Answer:

Question 17.
ln \(\frac{x}{3 y}\)
Answer:

Question 18.
ln \(\frac{6 x^{2}}{y^{4}}\)
Answer:

Question 19.
log₇ 5\(\sqrt{x}\)
Answer:

Question 20.
log₅ \(\sqrt[3]{x^{2} y}\)
Answer:

ERROR ANALYSIS In Exercises 21 and 22, describe and correct the error in expanding the logarithmic expression.
Question 21.

Answer:

Question 22.

Answer:

In Exercises 23–30, condense the logarithmic expression.
Question 23.
log₄ 7 − log₄ 10
Answer:

Question 24.
ln 12 − ln 4
Answer:

Question 25.
6 ln x+ 4 ln y
Answer:

Question 26.
2 log x+ log 11
ans;

Question 27.
log₅ 4 + \(\frac{1}{3}\) log₅ x
Answer:

Question 28.
6 ln 2 − 4 ln y
Answer:

Question 29.
5 ln 2 + 7 ln x + 4 ln y
Answer:

Question 30.
log₃ 4 + 2 log₃ \(\frac{1}{2}\) + log₃ x
Answer:

Question 31.
REASONING
Which of the following is not equivalent to log₃ \(\frac{y^{4}}{3 x}\)? Justify your answer.
A. 4 log₅ y − log₅ 3x
B. 4 log₅ y − log₅ 3 + log₅ x
C. 4 log₅ y− log₅ 3 − log₅ x
D. log₅ y⁴ − log₅ 3 − log₅ x
Answer:

Question 32.
REASONING
Which of the following equations is correct? Justify your answer.
A. log₇ x + 2 log₇ y = log₇ (x + y²)
B. 9 log x − 2 log y = log \(\frac{x^{9}}{y^{2}}\)
C. 5 log₄ x + 7 log₂ y = log₆x⁵y⁷
D. log₉x − 5 log₉ y = log₉ \(\frac{x}{5 y}\)
Answer:

In Exercises 33–40, use the change-of-base formula to evaluate the logarithm.
Question 33.
log₄ 7
Answer:

Question 34.
log₅ 13
Answer:

Question 35.
log₉ 15
Answer:

Question 36.
log₈ 22
Answer:

Question 37.
log₆ 17
Answer:

Question 38.
log₂ 28
Answer:

Question 39.
log₇ \(\frac{3}{16}\)
Answer:

Question 40.
log₃ \(\frac{9}{40}\)
Answer:

Question 41.
MAKING AN ARGUMENT
Your friend claims you can use the change-of-base formula to graph y = log₃x using a graphing calculator. Is your friend correct? Explain your reasoning.
Answer:

Question 42.
HOW DO YOU SEE IT?
Use the graph to determine the value of \(\frac{\log 8}{\log 2}\).

Answer:

MODELING WITH MATHEMATICS In Exercises 43 and 44, use the function L(I ) given in Example 6.
Question 43.
The blue whale can produce sound with an intensity that is 1 million times greater than the intensity of the loudest sound a human can make. Find the difference in the decibel levels of the sounds made by a blue whale and a human.

Answer:

Question 44.
The intensity of the sound of a certain television advertisement is 10 times greater than the intensity of the television program. By how many decibels does the loudness increase?

Answer:

Question 45.
REWRITING A FORMULA
Under certain conditions, the wind speed s (in knots) at an altitude of h meters above a grassy plain can be modeled by the function s(h) = 2 ln 100h.
a. By what amount does the wind speed increase when the altitude doubles?
b. Show that the given function can be written in terms of common logarithms as
s(h) = \(\frac{2}{\log e}\)(log h+ 2).
Answer:

Question 46.
THOUGHT PROVOKING
Determine whether the formula
log_b (M + N) = log_b M + log_bN
is true for all positive, real values of M, N, and b(with b≠ 1). Justify your answer.
Answer:

Question 47.
USING STRUCTURE
Use the properties of exponents to prove the change-of-base formula. (Hint: Let x= log_b a, y = log_b c, and z = log_b a.)
Answer:

Question 48.
CRITICAL THINKING
Describe three ways to transform the graph of f(x) = log x to obtain the graph of g(x) = log 100x − 1. Justify your answers.
Answer:

Maintaining Mathematical Proficiency

Solve the inequality by graphing.
Question 49.
x² − 4 > 0
Answer:

Question 50.
2(x − 6)² − 5 ≥ 37
Answer:

Question 51.
x² + 13x + 42 < 0
Answer:

Question 52.
−x² − 4x + 6 ≤ −6
Answer:

Solve the equation by graphing the related system of equations.
Question 53.
4x² − 3x − 6 = −x² + 5x + 3
Answer:

Question 54.
−(x + 3)(x − 2) = x² − 6x
Answer:

Question 55.
2x² − 4x − 5 = −(x + 3)² + 10
Answer:

Question 56.
−(x + 7)² + 5 = (x + 10)² − 3
Answer:

Lesson 6.6 Solving Exponential and Logarithmic Equations

Essential Question How can you solve exponential and logarithmic equations?

EXPLORATION 1

Solving Exponential and Logarithmic Equations
Work with a partner. Match each equation with the graph of its related system of equations. Explain your reasoning. Then use the graph to solve the equation.
a. e^x = 2
b. ln x = −1
c. 2^x = 3^-x
d. log₄x = 1
e. log₅ x = \(\frac{1}{2}\)
f. 4^x = 2

EXPLORATION 2

Solving Exponential and Logarithmic Equations
Work with a partner. Look back at the equations in Explorations 1(a) and 1(b). Suppose you want a more accurate way to solve the equations than using a graphical approach.
a. Show how you could use a numerical approach by creating a table. For instance, you might use a spreadsheet to solve the equations.

b. Show how you could use an analytical approach. For instance, you might try solving the equations by using the inverse properties of exponents and logarithms.

Communicate Your Answer

Question 3.
How can you solve exponential and logarithmic equations?
Answer:

Question 4.
Solve each equation using any method. Explain your choice of method.
a. 16^x = 2
b. 2^x = 4^2x+1
c. 2^x = 3^x+1
d. log x = \(\frac{1}{2}\)
e. ln x = 2
f. log₃ x = \(\frac{3}{2}\)
Answer:

Monitoring Progress

Solve the equation.
Question 1.
2^x = 5
Answer:

Question 2.
7^9x = 15
Answer:

Question 3.
4e^-0.3x − 7 = 13
Answer:

Question 4.
WHAT IF?
In Example 2, how long will it take to cool the stew to 100ºF when the room temperature is 75ºF?
Answer:

Solve the equation. Check for extraneous solutions.
Question 5.
ln(7x − 4) = ln(2x + 11)
Answer:

Question 6.
log₂(x − 6) = 5
Answer:

Question 7.
log 5x + log(x − 1) = 2
Answer:

Question 8.
log₄ (x + 12) + log₄ x = 3
Answer:

Solve the inequality.
Question 9.
e^x < 2
Answer:

Question 10.
10^2x-6 > 3
Answer:

Question 11.
log x + 9 < 45 Answer: Question 12. 2 ln x − 1 > 4
Answer:

Solving Exponential and Logarithmic Equations 6.6 Exercises

Vocabulary and Core Concept Check
Question 1.
COMPLETE THE SENTENCE
The equation 3^x-1 = 34 is an example of a(n) ___________ equation.
Answer:

Question 2.
WRITING
Compare the methods for solving exponential and logarithmic equations.
Answer:

Question 3.
WRITING
When do logarithmic equations have extraneous solutions?
Answer:

Question 4.
COMPLETE THE SENTENCE
If b is a positive real number other than 1, then b^x = b^y if and only if _________
Answer:

Monitoring Progress and Modeling with Mathematics

In Exercises 5–16, solve the equation.
Question 5.
7^3x+5 = 7^1-x
Answer:

Question 6.
e^2x = e^3x-1
Answer:

Question 7.
5^x-3 = 25^x-5
Answer:

Question 8.
6^2x-6 = 36^3x-5
Answer:

Question 9.
3^x = 7
Answer:

Question 10.
5^x = 33
Answer:

Question 11.
49^5x+2 = (\(\frac{1}{7}\))^11-x
Answer:

Question 12.
512^5x-1 = (\(\frac{1}{8}\))^-4-x
Answer:

Question 13.
7^5x = 12
Answer:

Question 14.
11^6x = 38
Answer:

Question 15.
3e^4x+9 = 15
Answer:

Question 16.
2e^2x − 7 = 5
Answer:

Question 17.
MODELING WITH MATHEMATICS
The length ℓ(in centimeters) of a scalloped hammerhead shark can be modeled by the function ℓ = 266 − 219^-0.05t
where t is the age (in years) of the shark. How old is a shark that is 175 centimeters long?

Answer:

Question 18.
MODELING WITH MATHEMATICS
One hundred grams of radium are stored in a container. The amount R(in grams) of radium present after t years can be modeled by R= 100e^−0.00043t. After how many years will only 5 grams of radium be present?
Answer:

In Exercises 19 and 20, use Newton’s Law of Cooling to solve the problem.
Question 19.
You are driving on a hot day when your car overheats and stops running. The car overheats at 280°F and can be driven again at 230°F. When it is 80°F outside, the cooling rate of the car is r = 0.0058. How long do you have to wait until you can continue driving?

Answer:

Question 20.
You cook a turkey until the internal temperature reaches 180°F. The turkey is placed on the table until the internal temperature reaches 100°F and it can be carved. When the room temperature is 72°F, the cooling rate of the turkey is r = 0.067. How long do you have to wait until you can carve the turkey?
Answer:

In Exercises 21–32, solve the equation.
Question 21.
ln(4x − 7) = ln(x + 11)
Answer:

Question 22.
ln(2x − 4) = ln(x + 6)
Answer:

Question 23.
log₂(3x − 4) = log₂ 5
Answer:

Question 24.
log(7x + 3) = log 38
Answer:

Question 25.
log₂(4x + 8) = 5
Answer:

Question 26.
log₃(2x + 1) = 2
Answer:

Question 27.
log₇(4x + 9) = 2
Answer:

Question 28.
log₅(5x + 10) = 4
Answer:

Question 29.
log(12x − 9) = log 3x
Answer:

Question 30.
log₆(5x + 9) = log₆ 6x
Answer:

Question 31.
log₂(x² − x − 6) = 2
Answer:

Question 32.
log₃(x² + 9x + 27) = 2
Answer:

In Exercises 33–40, solve the equation. Check for extraneous solutions.
Question 33.
log₂ x + log₂ (x − 2) = 3
Answer:

Question 34.
log₆ 3x + log₆ (x − 1) = 3
Answer:

Question 35.
ln x + ln(x + 3) = 4
Answer:

Question 36.
ln x + ln(x − 2) = 5
Answer:

Question 37.
log₃ 3x² + log₃ 3 = 2
Answer:

Question 38.
log₄ (−x) + log₄ (x + 10) = 2
Answer:

Question 39.
log₃(x − 9) + log₃(x − 3) = 2
Answer:

Question 40.
log₅ (x + 4) + log₅ (x + 1) = 2
Answer:

ERROR ANALYSIS In Exercises 41 and 42, describe and correct the error in solving the equation.
Question 41.

Answer:

Question 42.

Answer:

Question 43.
PROBLEM SOLVING
You deposit $100 in an account that pays 6% annual interest. How long will it take for the balance to reach $1000 for each frequency of compounding?
a. annual
b. quarterly
c. daily
d. continuously
Answer:

Question 44.
MODELING WITH MATHEMATICS
The apparent magnitude of a star is a measure of the brightness of the star as it appears to observers on Earth. The apparent magnitude M of the dimmest star that can be seen with a telescope is M = 5 log D+ 2, where D is the diameter (in millimeters) of the telescope’s objective lens. What is the diameter of the objective lens of a telescope that can reveal stars with a magnitude of 12?
Answer:

Question 45.
ANALYZING RELATIONSHIPS
Approximate the solution of each equation using the graph.

Answer:

Question 46.
MAKING AN ARGUMENT
Your friend states that a logarithmic equation cannot have a negative solution because logarithmic functions are not defined for negative numbers. Is your friend correct? Justify your answer.
Answer:

In Exercises 47–54, solve the inequality.
Question 47.
9^x > 54
Answer:

Question 48.
4^x ≤ 36
Answer:

Question 49.
ln x ≥ 3
Answer:

Question 50.
log₄ x< 4
Answer:

Question 51.
3^4x-5 < 8
Answer:

Question 52.
e^3x+4 > 11
Answer:

Question 53.
−3 log₅ x + 6 ≤ 9
Answer:

Question 54.
−4 log₅ x − 5 ≥ 3
Answer:

Question 55.
COMPARING METHODS
Solve log₅ x< 2 algebraically and graphically. Which method do you prefer? Explain your reasoning.
Answer:

Question 56.
PROBLEM SOLVING
You deposit $1000 in an account that pays 3.5% annual interest compounded monthly. When is your balance at least $1200? $3500?
Answer:

Question 57.
PROBLEM SOLVING
An investment that earns a rate of return r doubles in value in t years, where t = \(\frac{\ln 2}{\ln (1+r)}\) and r is expressed as a decimal. What rates of return will double the value of an investment in less than 10 years?
Answer:

Question 58.
PROBLEM SOLVING
Your family purchases a new car for $20,000. Its value decreases by 15% each year. During what interval does the car’s value exceed $10,000?
Answer:

USING TOOLS In Exercises 59–62, use a graphing calculator to solve the equation.
Question 59.
ln 2x = 3^-x+2
Answer:

Question 60.
log x = 7^-x
Answer:

Question 61.
log x = 3^x-3
Answer:

Question 62.
ln 2x = e^x-3
Answer:

Question 63.
REWRITING A FORMULA
A biologist can estimate the age of an African elephant by measuring the length of its footprint and using the equation ℓ = 45 − 25.7e^−0.09a, where ℓis the length (in centimeters) of the footprint and a is the age (in years).

a. Rewrite the equation, solving for a in terms of ℓ.
b. Use the equation in part (a) to find the ages of the elephants whose footprints are shown.
Answer:

Question 64.
HOW DO YOU SEE IT?
Use the graph to solve the inequality 4 ln x + 6 > 9. Explain your reasoning.

Answer:

Question 65.
OPEN-ENDED
Write an exponential equation that has a solution of x = 4. Then write a logarithmic equation that has a solution of x = −3.
Answer:

Question 66.
THOUGHT PROVOKING
Give examples of logarithmic or exponential equations that have one solution, two solutions, and no solutions.
Answer:

CRITICAL THINKING In Exercises 67–72, solve the equation.
Question 67.
2^x+3 = 5^3x-1
Answer:

Question 68.
10^3x-8 = 2^5-x
Answer:

Question 69.
log₃ (x − 6) = log₉ 2x
Answer:

Question 70.
log₄ x = log₈ 4x
Answer:

Question 71.
2^2x − 12 • 2^x + 32 = 0
Answer:

Question 72.
5^2x + 20 • 5^x − 125 = 0
Answer:

Question 73.
WRITING
In Exercises 67–70, you solved exponential and logarithmic equations with different bases. Describe general methods for solving such equations.
Answer:

Question 74.
PROBLEM SOLVING
When X-rays of a fixed wavelength strike a material x centimeters thick, the intensity I(x) of the X-rays transmitted through the material is given by I(x) = I₀e^−μx, where I0 is the initial intensity and μ is a value that depends on the type of material and the wavelength of the X-rays. The table shows the values of μ for various materials and X-rays of medium wavelength.

a. Find the thickness of aluminum shielding that reduces the intensity of X-rays to 30% of their initial intensity. (Hint: Find the value of x for which I(x) = 0.3I₀.)
b. Repeat part (a) for the copper shielding.
c. Repeat part (a) for the lead shielding.
d. Your dentist puts a lead apron on you before taking X-rays of your teeth to protect you from harmful radiation. Based on your results from parts (a)–(c), explain why lead is a better material to use than aluminum or copper.
Answer:

Maintaining Mathematical Proficiency

Write an equation in point-slope form of the line that passes through the given point and has the given slope.
Question 75.
(1, −2); m = 4
Answer:

Question 76.
(3, 2); m = −2
Answer:

Question 77.
(3, −8); m = − \(\frac{1}{3}\)
Answer:

Question 78.
(2, 5); m = 2
Answer:

Use finite differences to determine the degree of the polynomial function that fits the data. Then use technology to find the polynomial function.
Question 79.
(−3, −50), (−2, −13), (−1, 0), (0, 1), (1, 2), (2, 15), (3, 52), (4, 125)
Answer:

Question 80.
(−3, 139), (−2, 32), (−1, 1), (0, −2), (1, −1), (2, 4), (3, 37), (4, 146)
Answer:

Question 81.
(−3, −327), (−2, −84), (−1, −17), (0, −6), (1, −3), (2, −32), (3, −189), (4, −642)
Answer:

Lesson 6.7 Modeling with Exponential and Logarithmic Functions

Essential Question How can you recognize polynomial, exponential, and logarithmic models?

EXPLORATION 1

Recognizing Different Types of Models
Work with a partner. Match each type of model with the appropriate scatter plot. Use a regression program to find a model that fits the scatter plot.
a. linear (positive slope)
b. linear (negative slope)
c. quadratic
d. cubic
e. exponential
f. logarithmic

EXPLORATION 2

Exploring Gaussian and Logistic Models

Work with a partner. Two common types of functions that are related to exponential functions are given. Use a graphing calculator to graph each function. Then determine the domain, range, intercept, and asymptote(s) of the function.
a. Gaussian Function: f(x) = e^−x2
b. Logistic Function: f(x) = \(\frac{1}{1+e^{-x}}\)

Communicate Your Answer

Question 3.
How can you recognize polynomial, exponential, and logarithmic models?
Answer:

Question4 .
Use the Internet or some other reference to find real-life data that can be modeled using one of the types given in Exploration 1. Create a table and a scatter plot of the data. Then use a regression program to find a model that fits the data.
Answer:

Monitoring Progress

Determine the type of function represented by the table. Explain your reasoning.
Question 1.

Answer:

Question 2.

Answer:

Write an exponential function y = ab^x whose graph passes through the given points.
Question 3.
(2, 12), (3, 24)
Answer:

Question 4.
(1, 2), (3, 32)
Answer:

Question 5.
(2, 16), (5, 2)
Answer:

Question 6.
WHAT IF?
Repeat Examples 3 and 4 using the sales data from another store.

Answer:

Question 7.
Use a graphing calculator to find an exponential model for the data in Monitoring Progress Question 6.
Answer:

Question 8.
Use a graphing calculator to find a logarithmic model of the form p = a + b ln h for the data in Example 6. Explain why the result is an error message.
Answer:

Modeling with Exponential and Logarithmic Functions 6.7 Exercises

Vocabulary and Core Concept Check
Question 1.
COMPLETE THE SENTENCE
Given a set of more than two data pairs (x, y), you can decide whether a(n) __________ function fits the data well by making a scatter plot of the points (x, ln y).
Answer:

Question 2.
WRITING
Given a table of values, explain how you can determine whether an exponential function is a good model for a set of data pairs (x, y).
Answer:

Monitoring Progress and Modeling with Mathematics

In Exercises 3–6, determine the type of function represented by the table. Explain your reasoning.
Question 3.

Answer:

Question 4.

Answer:

Question 5.

Answer:

Question 6.

Answer:

In Exercises 7–16, write an exponential function y = ab^x whose graph passes through the given points.
Question 7.
(1, 3), (2, 12)
Answer:

Question 8.
(2, 24), (3, 144)
Answer:

Question 9.
(3, 1), (5, 4)
Answer:

Question 10.
(3, 27), (5, 243)
Answer:

Question 11.
(1, 2), (3, 50)
Answer:

Question 12.
(1, 40), (3, 640)
Answer:

Question 13.
(−1, 10), (4, 0.31)
Answer:

Question 14.
(2, 6.4), (5, 409.6)
Answer:

Question 15.

Answer:

Question 16.

Answer:

ERROR ANALYSIS In Exercises 17 and 18, describe and correct the error in determining the type of function represented by the data.
Question 17.

Answer:

Question 18.

Answer:

Question 19.
MODELING WITH MATHEMATICS
A store sells motorized scooters. The table shows the numbers y of scooters sold during the xth year that the store has been open. Write a function that models the data.

Answer:

Question 20.
MODELING WITH MATHEMATICS
The table shows the numbers y of visits to a website during the xth month. Write a function that models the data. Then use your model to predict the number of visits after 1 year.

Answer:

In Exercises 21–24, determine whether the data show an exponential relationship. Then write a function that models the data.
Question 21.

Answer:

Question 22.

Answer:

Question 23.

Answer:

Question 24.

Answer:

Question 25.
MODELING WITH MATHEMATICS
Your visual near point is the closest point at which your eyes can see an object distinctly. The diagram shows the near point y (in centimeters) at age x (in years). Create a scatter plot of the data pairs (x, ln y) to show that an exponential model should be a good fit for the original data pairs (x, y). Then write an exponential model for the original data.

Answer:

Question 26.
MODELING WITH MATHEMATICS
Use the data from Exercise 19. Create a scatter plot of the data pairs (x, ln y) to show that an exponential model should be a good fit for the original data pairs (x, y). Then write an exponential model for the original data.
Answer:

In Exercises 27–30, create a scatter plot of the points (x, ln y) to determine whether an exponential model fits the data. If so, find an exponential model for the data.
Question 27.

Answer:

Question 28.

Answer:

Question 29.

Answer:

Question 30.

Answer:

Question 31.
USING TOOLS
Use a graphing calculator to find an exponential model for the data in Exercise 19. Then use the model to predict the number of motorized scooters sold in the tenth year.
Answer:

Question 32.
USING TOOLS
A doctor measures an astronaut’s pulse rate y (in beats per minute) at various times x(in minutes) after the astronaut has finished exercising. The results are shown in the table. Use a graphing calculator to find an exponential model for the data. Then use the model to predict the astronaut’s pulse rate after 16 minutes.

Answer:

Question 33.
USING TOOLS
An object at a temperature of 160°C is removed from a furnace and placed in a room at 20°C. The table shows the temperatures d (in degrees Celsius) at selected times t (in hours) after the object was removed from the furnace. Use a graphing calculator to find a logarithmic model of the form t = a +b ln d that represents the data. Estimate how long it takes for the object to cool to 50°C.

Answer:

Question 34.
USING TOOLS
The f-stops on a camera control the amount of light that enters the camera. Let s be a measure of the amount of light that strikes the film and let f be the f-stop. The table shows several f-stops on a 35-millimeter camera. Use a graphing calculator to find a logarithmic model of the form s = a + b ln f that represents the data. Estimate the amount of light that strikes the film when f = 5.657.

Answer:

Question 35.
DRAWING CONCLUSIONS
The table shows the average weight (in kilograms) of an Atlantic cod that is x years old from the Gulf of Maine.

a. Show that an exponential model fits the data. Then find an exponential model for the data.
b. By what percent does the weight of an Atlantic cod increase each year in this period of time? Explain.
Answer:

Question 36.
HOW DO YOU SEE IT?
The graph shows a set of data points (x, ln y). Do the data pairs (x, y) fit an exponential pattern? Explain your reasoning.

Answer:

Question 37.
MAKING AN ARGUMENT
Your friend says it is possible to find a logarithmic model of the form d = a + b ln t for the data in Exercise 33. Is your friend correct? Explain.
Answer:

Question 38.
THOUGHT PROVOKING
Is it possible to write y as an exponential function of x? Explain your reasoning. (Assume p is positive.)

Answer:

Question 39.
CRITICAL THINKING
You plant a sunflower seedling in your garden. The height h (in centimeters) of the seedling after t weeks can be modeled by the logistic function
h(t) = \(\frac{256}{1+13 e^{-0.65 t}}\)
a. Find the time it takes the sunflower seedling to reach a height of 200 centimeters.
b. Use a graphing calculator to graph the function. Interpret the meaning of the asymptote in the context of this situation.
Answer:

Maintaining Mathematical Proficiency

Tell whether x and y are in a proportional relationship. Explain your reasoning.
Question 40.
y = \(\frac{x}{2}\)
Answer:

Question 41.
y = 3x − 12
Answer:

Question 42.
y = \(\frac{5}{x}\)
Answer:

Question 43.
y = −2x
Answer:

Identify the focus, directrix, and axis of symmetry of the parabola. Then graph the equation.
Question 44.
x = \(\frac{1}{8}\)y²
Answer:

Question 45.
y = 4x²
Answer:

Question 46.
x² = 3y
Answer:

Question 47.
y² = \(\frac{2}{5}\)x
Answer:

Exponential and Logarithmic Functions Performance Task: Measuring Natural Disasters

6.5–6.7 What Did You Learn?

Core Vocabulary
exponential equations, p. 334
logarithmic equations, p. 335

Core Concepts
Section 6.5
Properties of Logarithms, p. 328
Change-of-Base Formula, p. 329

Section 6.6
Property of Equality for Exponential Equations, p. 334
Property of Equality for Logarithmic Equations, p. 335

Section 6.7
Classifying Data, p. 342
Writing Exponential Functions, p. 343
Using Exponential and Logarithmic Regression, p. 345

Mathematical Practices
Question 1.
Explain how you used properties of logarithms to rewrite the function in part (b) of Exercise 45 on page 332.
Answer:

Question 2.
How can you use cases to analyze the argument given in Exercise 46 on page 339?
Answer:

Performance Task Measuring Natural Disasters
In 2005, an earthquake measuring 4.1 on the Richter scale barely shook the city of Ocotillo, California, leaving virtually no damage. But in 1906, an earthquake with an estimated 8.2 on the same scale devastated the city of San Francisco. Does twice the measurement on the Richter scale mean twice the intensity of the earthquake?
To explore the answers to these questions and more, go to BigIdeasMath.com.

Exponential and Logarithmic Functions Chapter Review

6.1 Exponential Growth and Decay Functions (pp. 295–302)

Tell whether the function represents exponential growth or exponential decay. Identify the percent increase or decrease. Then graph the function.
Question 1.
f (x) = (\(\frac{1}{3}\))^x
Answer:

Question 2.
y = 5^x
Answer:

Question 3.
f(x) = (0.2)^x
Answer:

Question 4.
You deposit $1500 in an account that pays 7% annual interest. Find the balance after 2 years when the interest is compounded daily.
Answer:

6.2 The Natural Base e (pp. 303–308)

Simplify the expression.
Question 5.
e⁴ • e¹¹
Answer:

Question 6.
\(\frac{20 e^{3}}{10 e^{6}}\)
Answer:

Question 7.
(−3e^-5x)²
Answer:

Tell whether the function represents exponential growth or exponential decay. Then graph the function.
Question 8.
f(x) = \(\frac{1}{3}\)e^x
Answer:

Question 9.
y = 6e^-x
Answer:

Question 10.
y = 3e^-0.75x
Answer:

6.3 Logarithms and Logarithmic Functions (pp. 309–316)

Evaluate the logarithm.
Question 11.
log₂ 8
Answer:

Question 12.
log₆ \(\frac{1}{36}\)
Answer:

Question 13.
log₂ 1
Answer:

Find the inverse of the function.
Question 14.
f(x) = 8^x
Answer:

Question 15.
y = ln(x − 4)
Answer:

Question 16.
y = log(x+ 9)
Answer:

Question 17.
Graph y = log_1/5 x.
Answer:

6.4 Transformations of Exponential and Logarithmic Functions (pp. 317–324)

Describe the transformation of f represented by g. Then graph each function.
Question 18.
f(x) = e^-x, g(x) = ^-5x − 8
Answer:

Question 19.
f(x) = log₄ x, g(x) = \(\frac{1}{2}\)log₄ (x + 5)
Answer:

Write a rule for g.
Question 20.
Let the graph of g be a vertical stretch by a factor of 3, followed by a translation 6 units left and 3 units up of the graph of f(x) = e^x.
Answer:

Question 21.
Let the graph of g be a translation 2 units down, followed by a reflection in the y-axis of the graph of f(x) = log x.
Answer:

6.5 Properties of Logarithms (pp. 327–332)

Expand or condense the logarithmic expression.
Question 22.
log₈ 3xy
Answer:

Question 23.
log 10x³y
Answer:

Question 24.
ln \(\frac{3 y}{x^{5}}\)
Answer:

Question 25.
3 log₇ 4 + log₇ 6
Answer:

Question 26.
log₂ 12 − 2 log₂ x
Answer:

Question 27.
2 ln x + 5 ln 2 − ln 8
Answer:

Use the change-of-base formula to evaluate the logarithm.
Question 28.
log₂ 10
Answer:

Question 29.
log₇ 9
Answer:

Question 30.
log₂₃ 42
Answer:

6.6 Solving Exponential and Logarithmic Equations (pp. 333–340)

Solve the equation. Check for extraneous solutions.
Question 31.
5^x = 8
Answer:

Question 32.
log₃ (2x − 5) = 2
Answer:

Question 33.
ln x + ln(x + 2) = 3
Answer:

Solve the inequality.
Question 34.
6^x > 12
Answer:

Question 35.
ln x ≤ 9
Answer:

Question 36.
e_4x-2 ≥ 16
Answer:

6.7 Modeling with Exponential and Logarithmic Functions (pp. 341–348)

Write an exponential model for the data pairs (x, y).
Question 37.
(3, 8), (5, 2)
Answer:

Question 38.

Answer:

Question 39.
A shoe store sells a new type of basketball shoe. The table shows the pairs sold s over time t (in weeks). Use a graphing calculator to find a logarithmic model of the form s = a +b ln t that represents the data. Estimate how many pairs of shoes are sold after 6 weeks.

Answer:

Exponential and Logarithmic Functions Chapter Test

Graph the equation. State the domain, range, and asymptote.
Question 1.
y = (\(\frac{1}{2}\))^x
Answer:

Question 2.
y = log_1/5 x
Answer:

Question 3.
y = 4e^-2x
Answer:

Describe the transformation of f represented by g. Then write a rule for g.
Question 4.
f(x) = log x

Answer:

Question 5.
f(x) = e^x

Answer:

Question 6.
f(x) = \(\frac{1}{4}\))^x

Answer:

Evaluate the logarithm. Use log₃ 4 ≈ 1.262 and log₃ 13 ≈ 2.335, if necessary.
Question 7.
log₂ 52
Answer:

Question 8.
log₃ \(\frac{13}{9}\)
Answer:

Question 9.
log₃ 16
Answer:

Question 10.
log₃ 8 + log₃ \(\frac{1}{2}\)
Answer:

Question 11.
Describe the similarities and differences in solving the equations 4^5x-2 = 16 and log₄(10x + 6) = 1. Then solve each equation.
Answer:

Question 12.
Without calculating, determine whether log₅11, \(\frac{\log 11}{\log 5}\), and \(\frac{\ln 11}{\ln 5}\) are equivalent expressions. Explain your reasoning.
Answer:

Question 13.
The amount y of oil collected by a petroleum company drilling on the U.S. continental shelf can be modeled by y = 12.263 ln x − 45.381, where y is measured in billions of barrels and x is the number of wells drilled. About how many barrels of oil would you expect to collect after drilling 1000 wells? Find the inverse function and describe the information you obtain from finding the inverse.
Answer:

Question 14.
The percent L of surface light that filters down through bodies of water can be modeled by the exponential function L(x) = 100e^kx, where k is a measure of the murkiness of the water and x is the depth (in meters) below the surface.

a. A recreational submersible is traveling in clear water with a k-value of about −0.02. Write a function that gives the percent of surface light that filters down through clear water as a function of depth.
b. Tell whether your function in part (a) represents exponential growth or exponential decay. Explain your reasoning.
c. Estimate the percent of surface light available at a depth of 40 meters.
Answer:

Question 15.
The table shows the values y (in dollars) of a new snowmobile after x years of ownership. Describe three different ways to find an exponential model that represents the data. Then write and use a model to find the year when the snowmobile is worth $2500.

Answer:

Exponential and Logarithmic Functions Cumulative Assessment

Question 1.
Select every value of b for the equation y = b^x that could result in the graph shown.

Answer:

Question 2.
Your friend claims more interest is earned when an account pays interest compounded continuously than when it pays interest compounded daily. Do you agree with your friend? Justify your answer.
Answer:

Question 3.
You are designing a rectangular picnic cooler with a length four times its width and height twice its width. The cooler has insulation that is 1 inch thick on each of the four sides and 2 inches thick on the top and bottom.

a. Let x represent the width of the cooler. Write a polynomial function T that gives the volume of the rectangular prism formed by the outer surfaces of the cooler.
b.Write a polynomial function C for the volume of the inside of the cooler.
c. Let I be a polynomial function that represents the volume of the insulation. How is I related to T and C?
d. Write I in standard form. What is the volume of the insulation when the width of the cooler is 8 inches?
Answer:

Question 4.
What is the solution to the logarithmic inequality −4 log₂ x ≥ −20?
A. x ≤ 32
B. 0 ≤ x ≤ 32
C. 0 < x ≤ 32
D. x ≥ 32
Answer:

Question 5.
Describe the transformation of f(x) = log₂ x represented by the graph of g.

Answer:

Question 6.
Let f(x) = 2x³ − 4x² + 8x− 1, g(x) = 2x − 3x⁴ − 6x³+ 5, and h(x) = −7 + x² + x. Order the following functions from least degree to greatest degree.
A. (f + g)(x)
B. (hg)(x)
C. (h − f)(x)
D. (fh)(x)
Answer:

Question 7.
Write an exponential model that represents each data set. Compare the two models.

Answer:

Question 8.
Choose a method to solve each quadratic equation. Explain your choice of method.
a. x² + 4x = 10
b. x² = −12
c. 4(x − 1)² = 6x + 2
d. x^x − 3x − 18 = 0
Answer:

Question 9.
At the annual pumpkin-tossing contest, contestants compete to see whose catapult will send pumpkins the longest distance. The table shows the horizontal distances y (in feet) a pumpkin travels when launched at different angles x (in degrees). Create a scatter plot of the data. Do the data show a linear, quadratic, or exponential relationship? Use technology to find a model for the data. Find the angle(s) at which a launched pumpkin travels 500 feet.

Answer:

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• Chapter 1 Understand Multiplication and Division
• Chapter 2 Multiplication Facts and Strategies
• Chapter 3 More Multiplication Facts and Strategies
• Chapter 4 Division Facts and Strategies
• Chapter 5 Patterns and Fluency
• Chapter 6 Relate Area to Multiplication
• Chapter 7 Round and Estimate Numbers
• Chapter 8 Add and Subtract Multi-Digit Numbers
• Chapter 9 Multiples and Problem Solving
• Chapter 10 Understand Fractions
• Chapter 11 Understand Fraction Equivalence and Comparison
• Chapter 12 Understand Time, Liquid Volume, and Mass
• Chapter 13 Classify Two-Dimensional Shapes
• Chapter 14 Represent and Interpret Data
• Chapter 15 Find Perimeter and Area

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Big Ideas Math Answers Grade 6 Chapter 1 Numerical Expressions and Factors: Before knowing the numerical expressions, Grade 6 students must be familiar with composite numbers, factor pairs, and prime numbers. Refer to the concept of vocabulary terms, subtracting and adding the fractions, mixed fractions in the below sections.

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Big Ideas Math Book 6th Grade Answer Key Chapter 1 Numerical Expressions and Factors

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Performance Task

• Numerical Expressions and Factors Steam Video/Performance Task
• Numerical Expressions and Factors Getting Ready for Chapter 1

Lesson 1: Powers and Exponents

• Lesson 1.1 Powers and Exponents
• Powers and Exponents Practice 1.1

Lesson 2: Order of Operations

• Lesson 1.2 Order of Operations
• Order of Operations Practice 1.2

Lesson 3: Prime Factorization

• Lesson 1.3 Prime Factorization
• Prime Factorization Practice 1.3

Lesson 4: Greatest Common Factor

• Lesson 1.4 Greatest Common Factor
• Greatest Common Factor Practice 1.4

Lesson 5: Least Common Multiple

• Lesson 1.5 Least Common Multiple
• Least Common Multiple Practice 1.5

Chapter: 1 – Numerical Expressions and Factors

• Numerical Expressions and Factors Connecting Concepts
• Numerical Expressions and Factors Chapter Review
• Numerical Expressions and Factors Practice Test
• Numerical Expressions and Factors Cumulative Practice

Numerical Expressions and Factors Steam Video/Performance Task

Filling Piñatas

Common factors can be used to make identical groups of objects. Can you think of any situations in which you would want to separate objects into equal groups? Are there any common factors that may be more useful than others? Can you think of any other ways to use common factors?
watch the STEAM Video “Filling Piñatas.” Then answer the following questions. The table below shows the numbers of party favors that Alex and Enid use to make piñatas.

Question 1.
When finding the number of identical piñatas that can be made, why is it helpful for Alex and Enid to list the factors of each number given in the table?

Answer: By using the list of the factors of all the numbers Alex and Enid can make identical groups of the objects.

Question 2.
You want to create 6 identical piñatas. How can you change the numbers of party favors in the table to make this happen? Can you do this without changing the total number of party favors?

Answer: You can change the number of party favors to create 6 identical pinatas.
There are 100 Mints. So divide it into two identical groups.
Change the number of mints to 50. And add 50 to new identical pinatas.

Performance Task

Setting the Table

After completing this chapter, you will be able to use the concepts you learned to answer the questions in the STEAM Video Performance Task. You will be asked to plan a fundraising event with the items below.
72 chairs
48 balloons
24 flowers
36 candles

You will find the greatest number of identical tables that can be prepared, and what will be in each centerpiece. When making arrangements for a party, should a party planner always use the greatest number of identical tables possible? Explain why or why not.

Answer:
72 chairs = 2 × 36
= 2 × 2 × 18
= 2 × 2 × 2 × 9
= 2 × 2 × 2 × 3 × 3
2, 2, 2, 3, 3
Therefore, 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72 are the factors of 72.
48 balloons = 2 × 24
= 2 × 2 × 12
= 2 × 2 × 2 × 6
= 2 × 2 × 2 × 2 × 3
The positive Integer factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24 and 48.
The factors of number 24 are 1, 2, 3, 4, 6, 8, 12, 24.
The factors of number 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36
The greatest number of identical tables possible are 1, 2, 3, 4, 6, 12.

Numerical Expressions and Factors Getting Ready for Chapter 1

Chapter Exploration

Work with a partner. In Exercises 1–3, use the table.

1. Cross out the multiples of 2 that are greater than 2. Do the same for 3, 5, and 7.
2. The numbers that are not crossed out are called prime numbers. The numbers that are crossed out are called composite numbers. In your own words, describe the characteristics of prime numbers and composite numbers.
3. MODELING REAL LIFE Work with a partner. Cicadas are insects that live underground and emerge from the ground after x or x + 4 years. Is it possible that both x and x +4 are prime? Give some examples.

Answer:

The numbers that are not crossed are 2, 3, 5, 7, 9, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59,61, 67, 71, 73, 79, 83, 89, 97.
These are not multiples of any numbers. So, the above numbers are the prime numbers.

Vocabulary

The following vocabulary terms are defined in this chapter. Think about what each term might mean and record your thoughts.

Answer:
i. First, we solve any operations inside of parentheses or brackets. Second, we solve any exponents. Third, we solve all multiplication and division from left to right. Fourth, we solve all addition and subtraction from left to right.
ii. “Factors” are numbers we can multiply together to get another number. When we find the factors of two or more numbers, and then find some factors are the same, then they are the “common factors”.
iii. A common multiple is a whole number that is a shared multiple of each set of numbers. The multiples that are common to two or more numbers are called the common multiples of those numbers. The smallest positive number is a multiple of two or more numbers.

Lesson 1.1 Powers and Exponents

Exploration 1
Writing Expressions Using Exponents
Work with a partner. Copy and complete the table.

Answer:

i. In your own words, describe what the two numbers in the expression 3⁵ mean.

Answer: 3⁵ means the number 3 repeats 5 times.
3 × 3 × 3 × 3 × 3 = 243

EXPLORATION 2
Using a Calculator to Find a Pattern

Work with a partner. Copy the diagram. Use a calculator to find each value. Write one digit of the value in each box. Describe the pattern in the digits of the values.

Answer:

1.1 Lesson

A power is a product of repeated factors. The base of a power is the repeated factor. The exponent of a power indicates the number of times the base is used as a factor.

Try It
Write the product as a power.
Question 1.
2 × 2 × 2

Answer: 2³  = 8
Two cubed or three to the two. Here 2 is repeated three times.

Question 2.

Answer: 46656 = 6⁶

Six to the six. Here 6 is repeated six times.

Question 3.
15 × 15 × 15 × 15

Answer: 56025 = 15⁴

15 to the power 4. Here 15 is repeated four times.

Question 4.

Answer: 1280000000 = 20⁷

20 to the power 7. Here 20 is repeated seven times.

Try It
Find the value of the power.

Question 5.
6³

Answer: 6 × 6 × 6 = 216
The value of the power 6³ is 216.

Question 6.
9²

Answer: 9 × 9 = 81
The value of the power 9² is 81

Question 7.
3⁴

Answer: 3 × 3 × 3 × 3
The value of the power 3⁴ is 81.

Question 8.
18²

Answer: 18 × 18
The value of the power 18² is 324.

Try It
Determine whether the number is a perfect square.

Question 9.
25

Answer: 5²
Yes, 25 is the perfect square.
A perfect square is a number, from a given number system, that can be expressed as the square of a number from the same number system.

Question 10.
2

Answer: 2 is not a perfect square. 2 cannot be expressed as the square of a number from the same number system.

Question 11.
99

Answer: 99 is not a perfect square. 99 cannot be expressed as the square of a number from the same number system.

Question 12.
36

Answer: 6²
36 is a perfect square
A perfect square is a number, from a given number system, that can be expressed as the square of a number from the same number system.

Self-Assessment for Concepts & Skills

Solve each exercise. Then rate your understanding of the success criteria in your journal.
FINDING VALUES OF POWERS
Find the value of the power.

Question 13.
8²

Answer: 8 × 8 = 64
The value of the power 8² is 64.

Question 14.
3⁵

Answer: 3 × 3 × 3 × 3 × 3 = 243
The value of the power 3⁵ is 243.

Question 15.
11³

Answer: 11 × 11 × 11 = 1331
The value of the power 11³ is 1331.

Question 16.
VOCABULARY
How are exponents and powers different?

Answer:
An expression that represents repeated multiplication of the same factor is called a power. The number 5 is called the base, and the number 2 is called the exponent. The exponent corresponds to the number of times the base is used as a factor.

Question 17.
VOCABULARY
Is 10 a perfect square? Is 100 a perfect square? Explain.

Answer: 10 is not a perfect square.
A perfect square is a number that is generated by multiplying two equal integers by each other.
100 is a perfect square. Because 10 × 10 = 100.

Question 18.
WHICH ONE DOESN’T BELONG?
Which one does not belong with the other three? Explain your reasoning.

Answer:
2⁴ = 2 × 2 × 2 × 2 = 16
3² = 3 × 3 = 9
3 + 3 + 3 + 3 = 3 × 4
5.5.5 = 125
The 3rd option does not belong to the other three expressions.

Self-Assessment for Problem Solving

Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 19.
A square solar panel has an area of 16 square feet. Write the area as a power. Then find the side lengths of the panel.

Answer: 4 feet

Explanation:
Given that,
A square solar panel has an area of 16 square feet.
A = s × s
16 = s²
4² = s²
s = 4
Thus the side length of the panel is 4 feet.

Question 20.
The four-square court shown is a square made up of four identical smaller squares. What is the area of the court?

Answer:
Given,
The four-square court shown is a square made up of four identical smaller squares.
The side of each square is 6 feet.
6 + 6 = 12 feet
The area of the court is 12 ft × 12 ft
A = 144 square feet
Thus the area of the court is 144 square feet.

Question 21.
DIG DEEPER!
Each face of a number cube is a square with a side length of 16 millimeters. What is the total area of all of the faces of the number cube?

Answer:
Given that,
Each face of a number cube is a square with a side length of 16 millimeters.
Area of the cube = 6 a²
A = 6 × 16 × 16
A = 1536 sq. mm

Powers and Exponents Practice 1.1

Review & Refresh

Multiply.

Question 1.
150 × 2

Answer: 300

Explanation:
Multiply the two numbers 150 and 2.
First multiply 2 with ones place 2 × 0 = 0
Next multiply with tens place 2 × 50 = 100
Next multiply with hundreds place 2 × 100 = 200
200 + 100 = 300

Question 2.
175 × 8

Answer: 1400

Explanation:
Multiply the two numbers 175 and 8.
First, multiply 2 with ones place 8 × 5 = 40
Next multiply with tens place 8 × 70= 560
Next multiply with hundreds place 8 × 100 = 800
800 + 560 + 40 = 1400

Question 3.
123 × 3

Answer: 369

Explanation:
Multiply the two numbers 123 and 3.
First multiply 2 with ones place 3 × 3 = 9
Next multiply with tens place 3 × 20 = 60
Next multiply with hundreds place 3 ×100 = 300
300 + 60 + 9 = 369

Question 4.
151 × 9

Answer: 1359

Explanation:
Multiply the two numbers 151 and 9.
First multiply 2 with ones place 9 × 1 = 9
Next multiply with tens place 9 × 50 = 450
Next multiply with hundreds place 9 × 100 = 900
900 + 450 + 9 = 1359

Write the sentence as a numerical expression.

Question 5.
Add 5 and 8, then multiply by 4.

Answer: The numerical expression for the above sentence is 5 + 8 × 4

Question 6.
Subtract 7 from 11, then divide by 2.

Answer: The numerical expression for the above sentence is 11 – 7 ÷ 2
Round the number to the indicated place value.

Question 7.
4.03785 to the tenths

Answer: The number 4.03785 nearest to the tenths is 4.0

Question 8.
12.89503 to the hundredths

Answer: The number 12.89503 nearest to the hundredths is 12.90

Complete the sentence.

Question 9.

Answer: 3

Explanation:
(1/10) × 30 = 30/10 = 3
The product of 1/10 and 30 is 3.

Question 10.

Answer: 20

Explanation:
(4/5) × 25 = 4 × 5 = 20
The product of 4/5 and 25 is 20.

Concepts, Skills, & Problem Solving

WRITING EXPRESSIONS USING EXPONENTS
Copy and complete the table. (See Exploration 1, p. 3.)

Answer:

WRITING EXPRESSIONS AS POWERS
Write the product as a power.

Question 15.
9 × 9

Answer: The exponential form of the given expression is 9²

Question 16.
13 × 13

Answer: The exponential form of the given expression is 13²

Question 17.
15 × 15 × 15

Answer: The exponential form of the given expression is 15³

Question 18.
2.2.2.2.2

Answer: The exponential form of the given expression is 2⁵

Question 19.
14 × 14 × 14

Answer: The exponential form of the given expression is 14³

Question 20.
8.8.8.8

Answer: The exponential form of the given expression is 8⁴

Question 21.
11 × 11 × 11 × 11 × 11

Answer: The exponential form of the given expression is 11⁵

Question 22.
7.7.7.7.7.7

Answer: The exponential form of the given expression is 7⁶

Question 23.
16.16.16.16

Answer: The exponential form of the given expression is 16⁴

Question 24.
43 × 43 × 43 × 43 × 43

Answer: The exponential form of the given expression is 43⁵

Question 25.
167 × 167 × 167

Answer: The exponential form of the given expression is 167³

Question 26.
245.245.245.245

Answer: The exponential form of the given expression is 245⁴

FINDING VALUES OF POWERS
Find the value of the power.

Question 27.
5²

Answer: The value of the powers 5² is 5 × 5 = 25

Question 28.
4³

Answer: The value of the powers 4³ is 4 × 4 × 4 = 64

Question 29.
6²

Answer: The value of the powers 6² is 6 × 6 = 36

Question 30.
1⁷

Answer: The value of the powers 1⁷ is 1 × 1 × 1 × 1 × 1 × 1 × 1 = 1

Question 31.
0³

Answer: The value of the powers 0³ is 0 × 0 × 0 = 0

Question 32.
8⁴

Answer: The value of the powers 8⁴ is 8 × 8 × 8 × 8 = 4096

Question 33.
2⁴

Answer: The value of the powers 2⁴ is 2 × 2 × 2 × 2 = 64

Question 34.
12²

Answer: The value of the powers 12² is 12 × 12 = 144

Question 35.
7³

Answer: The value of the powers 7³ is 7 × 7 × 7 = 343

Question 36.
5⁴

Answer: The value of the powers 5⁴ is 5 × 5 × 5 × 5 = 625

Question 37.
2⁵

Answer: The value of the powers 2⁵ is 2 × 2 × 2 × 2 × 2 = 32

Question 38.
14²

Answer: The value of the powers 14² is 14 × 14 = 196

USING TOOLS
Use a calculator to find the value of the power.

Question 39.
7⁶

Answer: 7 × 7 × 7 × 7 × 7 × 7 = 117649

Question 40.
4⁸

Answer: 4 × 4 × 4 × 4 × 4 × 4 × 4 × 4 = 256

Question 41.
12⁴

Answer: 12 × 12 × 12 × 12 = 20736

Question 42.
17⁵

Answer: 17 × 17 × 17 × 17 × 17 = 1419857

Question 43.
YOU BE THE TEACHER
Your friend finds the value of 8³. Is your friend correct? Explain your reasoning.

Answer: Your friend is incorrect
8³ is nothing but 8 repeats 3 times.
8³ = 8 × 8 × 8 = 512

IDENTIFYING PERFECT SQUARES
Determine whether the number is a perfect square.

Question 44.
8

Answer: 8 is not the perfect square. 8 cannot be expressed as the square of a number from the same number system.

Question 45.
4

Answer: 4 is a perfect square.
A perfect square is a number, from a given number system, that can be expressed as the square of a number from the same number system.

Question 46.
81

Answer: 81 perfect square
A perfect square is a number, from a given number system, that can be expressed as the square of a number from the same number system.

Question 47.
44

Answer: 44 is not the perfect square. 44 cannot be expressed as the square of a number from the same number system

Question 48.
49

Answer: 49 is a perfect square
A perfect square is a number, from a given number system, that can be expressed as the square of a number from the same number system.

Question 49.
125

Answer: 125 is not the perfect square. 125 cannot be expressed as the square of a number from the same number system

Question 50.
150

Answer: 150 is not the perfect square. 150 cannot be expressed as the square of a number from the same number system

Question 51.
144

Answer: 144 is the perfect square
A perfect square is a number, from a given number system, that can be expressed as the square of a number from the same number system.

Question 52.
MODELING REAL LIFE
On each square centimeter of a person’s skin, there are about 39² bacteria. How many bacteria does this expression represent?

Answer:
Given,
On each square centimeter of a person’s skin, there are about 39² bacteria.
39²  = 39 × 39 = 1521 centimeters
Thus the bacteria represents 1521 centimeters.

Question 53.
REPEATED REASONING
The smallest figurine in a gift shop is 2 inches tall. The height of each figurine is twice the height of the previous figurine. What is the height of the tallest figurine?

Answer:
Given that,
The smallest figurine in a gift shop is 2 inches tall. The height of each figurine is twice the height of the previous figurine.
The second figurine is twice that of the first figurine = 2 × 2 = 4 inches
The third figurine is twice that of the second figurine = 4 × 4 = 16 inches
The fourth figurine is twice that of the third figurine = 16 × 16 = 256 inches
Thus the height of the tallest figurine is 256 inches.

Question 54.
MODELING REAL LIFE
A square painting measures 2 meters on each side. What is the area of the painting in square centimeters?

Answer:
Given that,
A square painting measures 2 meters on each side.
Area of the square = s × s
A = 2 m × 2 m = 4 sq. meters
Thus the area of the painting in square centimeters is 4.

Question 55.
NUMBER SENSE
Write three powers that have values greater than 120 and less than 130.

Answer:
11² = 11(11) = 121; this is between 120 and 130.
5³ = 5(5)(5) = 25(5) = 125; this is between 120 and 130.
2⁷ = 2(2)(2)(2)(2)(2)(2) = 4(2)(2)(2)(2)(2) = 8(2)(2)(2)(2) = 16(2)(2)(2) = 32(2)(2) = 64(2) = 128; this is between 120 and 130.

Question 56.
DIG DEEPER!
A landscaper has 125 tiles to build a square patio. The patio must have an area of at least 80 square feet.

a. What are the possible arrangements for the patio?

Answer:
Given that a square patio of at least 80 square feet is to be built from 125 tiles of length 12 inches or 1 foot.
Since there are 125 tiles and the patio has a shape of a square of at least 80 square feet, then the possible dimensions of the patio are
9 ft × 9 ft = 81 ft
10 ft × 10 ft = 100 ft, and
11 ft × 11 ft = 121 ft.

b. How many tiles are not used in each arrangement?

Answer:
For a patio of dimensions, 9ft by 9ft, the number of tiles that will not be used is given by 125 – 81 = 44
For a patio of dimensions, 10ft by 10ft, the number of tiles that will not be used is given by 125 – 100 = 25
For a patio of dimensions, 11ft by 11ft, the number of tiles that will not be used is given by 125 – 121 = 4

Question 57.
PATTERNS
Copy and complete the table. Describe what happens to the value of the power as the exponent decreases. Use this pattern to find the value of 4⁰.

Answer:

4⁰ = 1
Thus the value of 4⁰ is 1.

Question 58.
REPEATED REASONING
How many blocks do you need to add to Square 6 to get Square 7? to Square 9 to get Square 10? to Square 19 to get Square 20? Explain.

Answer:
You need to add 14 blocks to get square 7. The square 7 contains 7 × 7 = 49 blocks
You need to add 32 blocks to get square 9. The square 9 contains 9 × 9 = 81 blocks
You need to add 19 blocks to get square 10. The square 10 contains 10 × 10 = 100 blocks
You need to add 261 blocks to get square 19. The square 19 contains 19 × 19 = 361 blocks
You need to add 39 blocks to get square 20. The square 20 contains 20 × 20 = 400 blocks

Lesson 1.2 Order of Operations

Order of Operations

EXPLORATION 1
Comparing Different Orders

Work with a partner. Find the value of each expression by using different orders of operations. Are your answers the same?

Answer:
The answers for all the expressions are not the same. The values of each expression will change if you change the order of operations.
a. 3 + 2 × 2
5 × 2 = 10
Multiply, then add
3 + 2 × 2
3 + 4 = 7
b. Subtract then multiply
18 – 3 × 3
15 × 3 = 45
Multiply, then subtract
18 – 3 × 3
18 – 9 = 9
c. Multiply, then subtract
8 × 8 – 2
64 – 2 = 62
Subtract, then Multiply
8 × 8 – 2
8 × 6 = 48
d. Multiply, then add
6 × 6 + 2
36 + 2 = 38
Add, then multiply
6 × 6 + 2
6 × 8 = 48

EXPLORATION 2
Determining Order of Operations
Work with a partner.
a. Scientific calculators use a standard order of operations when evaluating expressions. Why is a standard order of operations needed?

Answer: The 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.

b. Use a scientific calculator to evaluate each expression in Exploration 1. Enter each expression exactly as written. For each expression, which order of operations is correct?

Answer:
a. 3 + 2 × 2  – Multiply, then add
b. 18 – 3 × 3 – Multiply, then subtract
c. 8 × 8 – 2 – Multiply, then subtract
d. 6 × 6 + 2 – Multiply, then add

c. What order of operations should be used to evaluate 3 + 2², 18 − 3², 8² − 2, and 6² + 2?

Answer:
Solve the expressions by using the calculator.
a. 3 + 2 × 2
3 + 4 = 7
b. 18 – 3 × 3
18 – 9 = 9
c. 8 × 8 – 2
64 – 2 = 62
d. 6 × 6 + 2
36 + 2 = 38
d. Do 18 ÷ 3.3 and 18 ÷ 3² have the same value? Justify your answer.

Answer: No

Explanation:
18 ÷ 3.3
(18 ÷ 3) × 3
6 × 3 = 18
18 ÷ 3²  = 2
By using the calculator you can find the difference.
e. How does evaluating powers fit into the order of operations?

Answer:
When an expression has parentheses and powers, evaluate it in the following order: contents of parentheses, powers from left to right, multiplication and division from left to right, and addition and subtraction from left to right.

1.2 Lesson

A numerical expression is an expression that contains numbers and operations. To evaluate, or find the value of, a numerical expression, use a set of rules called the order of operations.
Key Idea
order of operations

1. Perform operations in grouping symbols.
2. Evaluate numbers with exponents.
3. Multiply and divide from left to right.
4. Add and subtract from left to right.

Try It
a. Evaluate the expression.

Question 1.
7.5 + 3

Answer: 56

Explanation:
You have to evaluate the expression from left to right.
7(5 + 3) = 7 × 8
= 56

Question 2.
(28 – 20) ÷ 4

Answer: 2

Explanation:
You have to evaluate the expression from left to right.
28 – 20 = 8
8 ÷ 4 = 2

Question 3.
[6 + (15 – 10)] × 5

Answer: 55

Explanation:
You have to evaluate the expression from left to right.
[6 + (15 – 10)] × 5
[6 + 5] × 5
11 × 5 = 55

Try It
Evaluate the expression.

Question 4.
6 + 2⁴ – 1

Answer: 21

Explanation:
You have to evaluate the expression from left to right.
6 + 2⁴ – 1
6 + (16 – 1)
6 + 15 = 21
6 + 2⁴ – 1 = 21

Question 5.
4.3² + 18 – 9

Answer: 45

Explanation:
You have to evaluate the expression from left to right.
4.3² + (18 – 9)
4.3² + 9
4 × 9 + 9
36 + 9 = 45

Question 6.
16 + (5² – 7) ÷ 3

Answer:

Explanation:
You have to evaluate the expression from left to right.
16 + (5² – 7) ÷ 3
16 + (25 – 7) ÷ 3
16 + (18) ÷ 3
16 + (18 ÷ 3)
16 + 6 = 22

Thee symbols × and . are used to indicate multiplication. You can also use parentheses to indicate multiplication. For example, 3(2 +7) is the same as 3 × (2 + 7).

Try It
Evaluate the expression.

Question 7.
50 + 6(12 ÷ 4) – 8²

Answer: 4

Explanation:
You have to evaluate the expression from left to right.
50 + 6(12 ÷ 4) – 8²
50 + 6(3) – 8²
50 + 18 – 8²
50 + 18 – 64
68 – 64
4

Question 8.

Answer: 24

Explanation:
You have to evaluate the expression from left to right.
5² – 1/5 (10 – 5)
5² – 1/5 (5)
5² – 1
25 – 1
24

Question 9.

Answer:

Explanation:
You have to evaluate the expression from left to right.
8(2+5) = 8 × 7
(8 × 7)/7 = 8

Self-Assessment for Concepts & Skills

Solve each exercise. Then rate your understanding of the success criteria in your journal.

USING ORDER OF OPERATIONS
Evaluate the expression.

Question 10.
7 + 2.4

Answer: 15

Explanation:
You have to evaluate the expression from left to right.
7 + 2 × 4 = 7 + 8 = 15

Question 11.
8 ÷ 4 × 2

Answer: 4

Explanation:
You have to evaluate the expression from left to right.
8 ÷ 4 = 2
2 × 2 = 4

Question 12.
3(5 + 1) ÷ 3²

Answer: 2

Explanation:
You have to evaluate the expression from left to right.
3(5 + 1) ÷ 3²
3 × 6 ÷ 3²
18 ÷ 9 =2

Question 13.
WRITING
Why does 12 − 8 ÷ 2 = 2?

Answer:
12 − 8 ÷ 2
4 ÷ 2 = 2

Question 14.
REASONING
Describe the steps in evaluating the expression 8 ÷ (6 − 4) + 3².

Answer:
8 ÷ (6 − 4) + 3²
8 ÷ 2 + 3²
4 + 9 = 13

Question 15.
WHICH ONE DOESN’T BELONG?
Which expression does not belong with the other three? Explain your reasoning.

Answer: (5² – 8) × 2 does not belong to the other three. Because the order of operations and expressions are different for the fourth option.

Self-Assessment for Problem Solving

Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 16.
A square plot of land has side lengths of 40 meters. An archaeologist divides the land into 64 equal parts. What is the area of each part?

Answer:
Given that,
A square plot of land has side lengths of 40 meters.
An archaeologist divides the land into 64 equal parts.
Side of the square field = 40m
Area of the square field = s × s
A = 40m × 40m
A = 1600 sq.m
Area of each part of the square = 1600/64 = 25 sq.m

Question 17.
A glass block window is made of two different-sized glass squares. The window has side lengths of 40 inches. The large glass squares have side lengths of 10 inches. Find the total area of the small glass squares.

Answer:
Given,
A glass block window is made of two different-sized glass squares.
The window has side lengths of 40 inches. The large glass squares have side lengths of 10 inches.
40 × 10 = 400

Question 18.
DIG DEEPER!
A square vegetable garden has side lengths of 12 feet. You plant flowers in the center portion as shown. You divide the remaining space into 4 equal sections and plant tomatoes, onions, zucchini, and peppers. What is the area of the onion section?

Answer:
A square vegetable garden has side lengths of 12 feet.
You plant flowers in the center portion of the garden, a square that has side lengths of 4 feet.
You divide the remaining space into 4 equal sections and plant tomatoes, onions, zucchini, and peppers.
Given that,
→ side of the square vegetable garden is = 12 feet.
So,
→ Area of square vegetable garden = (side)² = (12)² = 144 feet².
Now, given that, inside this area, there is a square of side 4 feet reserved for flowers.
So,
→ The area of the flower section = (side)² = (4)² = 16 feet².
Therefore,
→ The rest of the garden that is intended for vegetables is = The total garden area – The flower section area = 144 – 16 = 128 feet².
Now, this remaining area is to be divided into four equal sections.
So,
→ The area of the onion section = (1/4) of remaining area = (1/4) × 128 = 32 feet².

Order of Operations Practice 1.2

Review & Refresh

Write the product as a power.

Question 1.
11 × 11 × 11 × 11

Answer: The exponent for the product 11 × 11 × 11 × 11 is 11⁴

Question 2.
13 × 13 × 13 × 13 × 13

Answer: The exponent for the product 13 × 13 × 13 × 13 × 13 is 13⁵
Find the missing dimension of the rectangular prism.

Question 3.

Answer:
Given that,
l = 6 in.
b = 4 in.
h = ?
v = 192 cu. in
Volume of the rectangular prism = lbh
192 = 6 × 4 × h
h = 192/24
h = 8
Thus the height of the rectangular prism is 8 inches.

Question 4.

Answer:
Given that,
h = 9m
b = 3m
v = 135 cu. m
l = ?
The volume of the rectangular prism = lbh
135 = l × 3 × 9
135 = l × 27
l = 5m

Tell whether the number is prime or composite.

Question 5.
9

Answer: Composite Number
A natural number greater than 1 that is not prime is called a composite number.

Question 6.
11

Answer: Prime Number
A prime number is a natural number greater than 1 that is not a product of two smaller natural numbers.

Question 7.
23

Answer: Prime Number
A prime number is a natural number greater than 1 that is not a product of two smaller natural numbers.

Concepts, Skills, & Problem Solving

COMPARING DIFFERENT ORDERS
Find the value of the expression by using different orders of operations. Are your answers the same? (See Exploration 1, p. 9.)

Question 8.

Answer:
4 + 6 × 6
10 × 6 = 60
4 + 6 × 6
4 + 36 = 40

Question 9.

Answer:
5 × 5 – 3
5 × 2 = 10
5 × 5 – 3
25 – 3 = 22

USING ORDER OF OPERATIONS
Evaluate the expression.

Question 10.
5 + 18 ÷ 6

Answer: 8

Explanation:
First, divide then divide.
5 + 3 = 8

Question 11.
(11 – 3) ÷ 2 + 1

Answer:

Explanation:
First, subtract and divide.
(11 – 3) ÷ 2 + 1
8 ÷ 2 + 1
4 + 1 = 5

Question 12.
45 ÷ 9 × 2

Answer: 10

Explanation:
The first divide then multiply.
45 ÷ 9 × 2
5 × 2 = 10

Question 13.
6² – 3.4

Answer: 24

Explanation:
Multiply then subtract
6² – 3.4
36 – 12
24

Question 14.
42 ÷ (15 – 2³)

Answer: 6

Explanation:
Subtract then divide.
42 ÷ (15 – 8)
42 ÷ 7
6

Question 15.
4².2 + 8.7

Answer: 88

Explanation:
Multiply then add
4².2 + 8.7
16 × 2 + 56
32 + 56 = 88

Question 16.
(5² – 2) × 1⁵ + 4

Answer: 27

Explanation:
(5² – 2) × 1⁵ + 4
(25 – 2) × 1 + 4
Add, subtract then multiply
23 + 4 = 27

Question 17.
4 + 2 × 3² – 9

Answer: 13

Explanation:
4 + 2 × 3² – 9
4 + 18 – 9
4 + 9 = 13

Question 18.
8 ÷ 2 × 3 + 4² ÷ 4

Answer: 16

Explanation:
8 ÷ 2 × 3 + 4² ÷ 4
(4 × 3) + (16 ÷ 4)
12 + 4
16

Question 19.
3² + 12 ÷ (6 – 3) × 8

Answer: 41

Explanation:
3² + 12 ÷ (6 – 3) × 8
9 + (12 ÷ (6 – 3)) × 8
9 + (12 ÷ 3) × 8
9 + 4 × 8
9 + 32
41

Question 20.
(10 + 4) ÷ (26 – 19)

Answer: 2

Explanation:
Add, subtract then divide
(10 + 4) ÷ (26 – 19)
14 ÷ 7
2

Question 21.
(5² – 4).2 – 18

Answer: 24

Explanation:
((5² – 4).2) – 18
((25 – 4) × 2) – 18
(21 × 2) – 18
42 – 18
24

Question 22.
2 × [(16 – 8) × 2]

Answer: 32

Explanation:
2 × [(16 – 8) × 2]
2 × [8 × 2]
2 × 16
32

Question 23.
12 + 8 × 3³ – 24

Answer: 204

Explanation:
12 + 8 × 3³ – 24
12 + (8 × 27) – 24
12 + 216 – 24
12 + 192 = 204

Question 24.
6² ÷ [(2 + 4) × 2³]

Answer: 48

Explanation:
6² ÷ [(2 + 4) × 2³]
36 ÷ [(2 + 4) × 2³]
36 ÷ 6 × 8
6 × 8
48

YOU BE THE TEACHER
Your friend evaluates the expression. Is your friend correct? Explain your reasoning.

Question 25.

Answer: Your friend is incorrect.
9 + 3 × 3²
9 + (27)
36

Question 26.

Answer:
19 – 6 + 12
13 + 12
25

Question 27.
PROBLEM SOLVING
You need to read 20 poems in 5 days for an English project. Each poem is 2 pages long. Evaluate the expression 20 × 2 ÷ 5 to find how many pages you need to read each day.

Answer:
Given,
You need to read 20 poems in 5 days for an English project. Each poem is 2 pages long.
20 × 2 ÷ 5
40 ÷ 5 = 8
Thus you need to read 8 pages each day.

USING ORDER OF OPERATIONS
Evaluate the expression.

Question 28.
12 – 2(7 – 4)

Answer:
12 -(2 × (7 – 4))
12 – (2 × 3)
12 – 6 = 6

Question 29.
4(3 + 5) – 3(6 -2)

Answer:
4(3 + 5) – 3(6 -2)
4 × 8 – 3 × 4
32 – 12
20

Question 30.

Answer:
6 + 1/4 (12 -8)
6 + 1/4(4)
6 + 1
7

Question 31.
9² – 8(6 + 2)

Answer:
81 – (8(6 + 2))
81 – (8 × 8)
81 – 64
17

Question 32.
4(3 – 1)³ + 7(6) – 5²

Answer:
4(3 – 1)³ + 7(6) – 5²
4(2)³ + 7(6) – 5²
4 × 8 + 42 – 25
32 + 42 – 25 = 49

Question 33.

Answer:
8[(1 1/6 + 5/6) ÷ 4]
[8[7/6 + 5/6] ÷ 4]
8[12/6] ÷ 4
8[2 ÷ 4]
8(1/2)
4

Question 34.

Answer:
49 – 2((11-3)/8)
49 – 2 (8/8)
49 – 2
47

Question 35.
8(7.3 + 3.7 – 8) ÷ 2

Answer:
8(7.3 + 3.7 – 8) ÷ 2
(8(7.3 + 3.7 – 8)) ÷ 2
8 (11 – 8) ÷ 2
8 × 3 ÷ 2
24 ÷ 2
12

Question 36.
2⁴(5.2 – 3.2) ÷ 4

Answer:
2⁴(5.2 – 3.2) ÷ 4
16 (5.2 – 3.2) ÷ 4
16 (2) ÷ 4
32 ÷ 4
8

Question 37.

Answer:
36(3+5)/4
36 × 8/4
36 × 2
72

Question 38.

Answer:
(144 – 24 + 1)/121
121/121
1

Question 39.

Answer:
26 ÷ 2 + 5 = 18
18/6 = 3

Question 40.
PROBLEM SOLVING
Before a show, there are 8 people in a theater. Five groups of 4 people enter, and then three groups of 2 people leave. Evaluate the expression 8 + 5(4) − 3(2) to find how many people are in the theater.

Answer:
Given,
Before a show, there are 8 people in a theater. Five groups of 4 people enter, and then three groups of 2 people leave.
8 + (5 × 4) – (3 × 2)
8 + 20 – 6
28 – 6
22

Question 41.
MODELING REAL LIFE
The front door of a house is painted white and blue. Each window is a square with a side length of 7 inches. What is the area of the door that is painted blue?

Answer:
Given,
The front door of a house is painted white and blue. Each window is a square with a side length of 7 inches.
Area of the square = s × s
A = 7 in × 7 in
A = 49 sq. inches
Therefore the area of the door that is painted blue is 49 sq. inches.

Question 42.
PROBLEM SOLVING
You buy 6 notebooks, 10 folders, 1 pack of pencils, and 1 lunch box for school. After using a $10 gift card, how much do you owe? Explain how you solved the problem.

Answer:
Given,
You buy 6 notebooks, 10 folders, 1 pack of pencils, and 1 lunch box for school.
Cost of 1 notebook = $2
6 notebooks = 6 × $2 = $12
Cost of 1 folder = $1
10 folders = 10 × $1 = $10
Cost of 1 pack of pencils = $3
Cost of 1 lunch box = $8
So the total cost is $11 + $10 + $3 + $8 = $31
You used $10 gift card.
$31 – $10 = $21
Thus you ow $21.

Question 43.
OPEN-ENDED
Use all four operations and at least one exponent to write an expression that has a value of 100.

Answer: You need to use +, -, ×, ÷  operations to write the expressions that have the value of 100.
(34 – 1) × 3 + 3² ÷ 9 = 100

Question 44.

REPEATED REASONING
A Petri dish contains 35 cells. Every day, each cell in the Petri dish divides into 2 cells in a process called mitosis. How many cells are there after 14 days? Justify your answer.

Answer:
Given,
A Petri dish contains 35 cells. Every day, each cell in the Petri dish divides into 2 cells in a process called mitosis
35 ÷ 2 = 17.5
1 day = 0.5 + 0.5 cells
14 days = 14 × 1 = 14 cells
17.5 – 14 = 3.5
Thus there are 3.5 cells after 14 days.

Question 45.
REASONING
Two groups collect litter along the side of a road. It takes each group 5 minutes to clean up a 200-yard section. How long does it take both groups working together to clean up 2 miles? Explain how you solved the problem.

Answer:
Given,
Two groups collect litter along the side of a road. It takes each group 5 minutes to clean up a 200-yard section.
To convert 2 miles to yards, you have to multiply 2 by 1760, because 1 mile equals to 1760 yards:
2 × 1760 = 3520 yards.
If you would like to know how long does it take to clean up 2 miles, you can calculate this using the following steps:
5 × 3520 = 200 × x
17600 = 200 × x /200
x = 17600 / 200
x = 88 minutes

Question 46.
NUMBER SENSE
Copy each statement. Insert +, −, ×, or ÷ symbols to make each statement true.

Answer:
You can find the value by using the calculator by inserting the suitable operations.

Lesson 1.3 Prime Factorization

Prime Factorization

EXPLORATION 1
Rewriting Numbers as Products of Factors

Work with a partner. Two students use factor trees to write 108 as a product of factors, as shown below.

a. Without using 1 as a factor, can you write 108 as a product with more factors than each student used? Justify your answer. Math Practice

Answer: Yes you can find the factors by using the prime factorization.
108 = 2 × 54
= 2 × 2 × 27
= 2 × 2 × 3 × 9
= 2 × 2 × 3 × 3 × 3

b. Use factor trees to write 80, 162, and 300 as products of as many factors as possible. Do not use 1 as a factor.

Answer:
80 = 2 × 40
= 2 × 2 × 20
= 2 × 2 × 2 × 10
= 2 × 2 × 2 × 2 × 5
162 = 2 × 81
= 2 × 3 × 27
= 2 × 3 × 3 × 9
= 2 × 3 × 3 × 3 × 3
300 = 2 × 150
= 2 × 2 × 75
= 2 × 2 × 3 × 25
= 2 × 2 × 3 × 5 × 5

c. Compare your results in parts (a) and (b) with other groups. For each number, identify the product with the greatest number of factors. What do these factors have in common?

Answer: 300 contains the greatest number of factors. (2, 2, 3, 5, 5)

1.3 Lesson

Because 2 is a factor of 10 and 2 . 5 =10, 5 is also a factor of 10. The pair 2, 5 is called a factor pair of 10.

Try It
List the factor pairs of the number.

Question 1.
18

Answer: The factor pairs of 18 are 1, 2, 3, 6, 9, 18

Explanation:
1 × 18 = 18
2 × 9 = 18
3 × 6 = 18
6 × 3 = 18
9 × 2 = 18
18 × 1 = 18

Question 2.
24

Answer: The factor pairs of 1, 2, 3, 4, 6, 8, 12, 24

Explanation:
1 × 24 = 24
2 × 12 = 24
3 × 8 = 24
4 × 6 = 24
6 × 4 = 24
8 × 3 = 24
12 × 2 = 24
24 × 1 = 24

Question 3.
51

Answer: The factor pairs of 1, 3, 17, 51

Explanation:
1 × 51 = 51
3 × 17 = 51
17 × 3 = 51
51 × 1 = 15

Question 4.
WHAT IF?
The woodwinds section of the marching band has 38 members. Which has more possible arrangements, the brass section or the woodwinds section? Explain.

Answer: Brass section. 38 has only two-factor pairs.
38 = 1 × 38
= 2 × 19

Key Idea

Prime Factorization
The prime factorization of a composite number is the number written as a product of its prime factors.
You can use factor pairs and a factor tree to help find the prime factorization of a number. The factor tree is complete when only prime factors appear in the product. A factor tree for 60 is shown.

Try It
Write the prime factorization of the number.

Question 5.
20

Answer:
The Prime Factorization is:
2 x 2 x 5
In Exponential Form:
2² x 51
CSV Format:
2, 2, 5

Question 6.
88

Answer:
88 = 2 × 44
= 2 × 2 × 22
= 2 × 2 × 2 × 11
The Prime Factorization is: 2 × 2 × 2 × 11

Question 7.
90

Answer:
90 = 2 × 45
= 2 × 3 × 15
= 2 × 3 × 3 × 5
The Prime Factorization is: 2 × 3 × 3 × 5

Question 8.
462

Answer:
= 2 × 231
= 2 × 3 × 77
= 2 × 3 × 7 × 11
The Prime Factorization is: 2 × 3 × 7 × 11

Self-Assessment for Concepts & Skills

Solve each exercise. Then rate your understanding of the success criteria in your journal.

WRITING A PRIME FACTORIZATION
Write the prime factorization of the number.

Question 9.
14

Answer:
14 = 2 × 7
The Prime Factorization is: 2 × 7

Question 10.
86

Answer:
86 = 2 × 43
The Prime Factorization is: 2 × 43

Question 11.
40

Answer:
40 = 2 × 20
= 2 × 2 × 10
= 2 × 2 × 2 × 5
The Prime Factorization is: 2 × 2 × 2 × 5

Question 12.
516

Answer:
516 = 2 × 258
= 2 × 2 × 129
= 2 × 2 × 3 × 43
The Prime Factorization is: 2 × 2 × 3 × 43

Question 13.
WRITING
Explain the difference between prime numbers and composite numbers.

Answer:
A prime number is a number that has exactly two factors i.e. ‘1’ and the number itself. A composite number has more than two factors, which means apart from getting divided by number 1 and itself, it can also be divided by at least one integer or number.

Question 14.
STRUCTURE
Your friend lists the following factor pairs and concludes that there are 6 factor pairs of 12. Explain why your friend is incorrect.

Answer: Your friend is incorrect. Because there are 5-factor pairs of 12.
The factor pairs of 12 are
1 × 12 =12
2 × 6 = 12
3 × 4 = 12
4 × 3 = 12
6 × 2 = 12

Question 15.
WHICH ONE DOESN’T BELONG?
Which factor pair does not belong with the other three? Explain your reasoning.

Answer:
2, 28 = 2 × 28 = 56
4, 14 = 4 × 14 = 56
6, 9 = 6 × 9 = 54
7, 8 = 7 × 56
By this we can say that 6, 9 does not belong to the other three expressions.

Self-Assessment for Problem Solving

Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 16.
A group of 20 friends plays a card game. The game can be played with 2 or more teams of equal size. Each team must have atleast 2 members. List the possible numbers and sizes of teams.

Answer:
Given,
A group of 20 friends plays a card game. The game can be played with 2 or more teams of equal size. Each team must have at least 2 members.
20 = 1 × 20
2 × 10
4 × 5
5 × 4
10 × 2
Thus there are 5 possible numbers and size of teams.

Question 17.

You arrange 150 chairs in rows for a school play. You want each row to have the same number of chairs. How many possible arrangements are there? Are all of the possible arrangements appropriate for the play? Explain.

Answer:
You arrange 150 chairs in rows for a school play. You want each row to have the same number of chairs.
150 = 1, 2, 3, 5, 6, 10, 15, 25, 30, 50, 75, 150
Thus there are 12 possible arrangements appropriate for the play.

Question 18.
What is the least perfect square that is a factor of 4536? What is the greatest perfect square that is a factor of 4536?

Answer: least perfect square that is a factor of 4536

Explanation:
What is the last number of 4,536? It is this number: 4536. The answer is 6. Is 6 in the list of numbers that are never perfect squares (2, 3, 7 or 8)?
Answer: No, 6 is not in the list of numbers that are never perfect squares. Let’s continue to the next step.
Step 2:
We now need to obtain the digital root of the number. Here’s how you do it:
Split the number up and add each digit together:
4 + 5 + 3 + 6 = 18
If the answer is more than one digit, you would add each digit of the answer together again:
1 + 8 = 9
1 x 4,5362 x 2,2683 x 1,5124 x 1,1346 x 7567 x 6488 x 5679 x 50412 x 37814 x 32418 x 25221 x 21624 x 18927 x 16828 x 16236 x 12642 x 10854 x 8456 x 8163 x 72
We’re looking for a factor combination with equal numbers for X and Y (like 3×3) above. Notice there isn’t an equal factor combination, that when multiplied together, produce the number 4,536. That means 4,536 is NOT a perfect square.

Question 19.
DIG DEEPER!
The prime factorization of a number is 2⁴ × 3⁴ × 5⁴ × 7². Is the number a perfect square? Explain your reasoning.

Answer:
The prime factorization of a number is 2⁴ × 3⁴ × 5⁴ × 7².
16 × 81 × 625 × 49 = 39690000
Yes, 39690000 is a perfect square.

Prime Factorization Practice 1.3

Review & Refresh

Evaluate the expression.

Question 1.
2 + 4²(5 – 3)

Answer: 34

Explanation:
2 + 4²(5 – 3)
2 + 4²(2)
2 + 16(2) = 2 + 32 = 34

Question 2.
2³ + 4 × 3²

Answer: 44

Explanation:
2³ + 4 × 3²
8 + 4 × 9
8 + 36
44

Question 3.

Answer:

Explanation:
9 × 5 – 16(5/2 – 1/2)
9 × 5 – 16(4/2)
45 – 16(2)
45 – 32
13

Plot the points in a coordinate plane. Draw a line segment connecting the points.

Question 4.
(1, 1) and (4, 3)

Answer:

Question 5.
(2, 3) and (5, 9)

Answer:

Question 6.
(2, 5) and (4, 8)

Answer:

Use the Distributive Property to find the quotient. Justify your answer.

Question 7.
408 ÷ 4

Answer: 120
Write 408 as 204 and 204
204 ÷ 4 = 51
204 ÷ 4 = 51
51 + 51 = 102
408 ÷ 4 = 102

Question 8.
628 ÷ 2

Answer: 314
608 can be written as 314 and 314
314 ÷ 2 = 157
314 ÷ 2 = 157
157 + 157 = 314
628 ÷ 2 = 314

Question 9.
969 ÷ 3

Answer: 323
969 can be written as 900 and 69
900 ÷ 3 = 300
69 ÷ 3 = 23
300 + 23 = 323
969 ÷ 3 = 323

Classify the triangle in as many ways as possible.

Question 10.

Answer: Acute
Acute angles measure less than 90 degrees. Right angles measure 90 degrees. Obtuse angles measure more than 90 degrees.

Question 11.

Answer: Obtuse
An obtuse angle has a measurement greater than 90 degrees but less than 180 degrees. However, A reflex angle measures more than 180 degrees but less than 360 degrees.

Question 12.

Answer: Right angle
In geometry and trigonometry, a right angle is an angle of exactly 90° (degrees), corresponding to a quarter turn. If a ray is placed so that its endpoint is on a line and the adjacent angles are equal, then they are right angles.

Concepts, Skills, & Problem Solving

REWRITING A NUMBER
Write the number as a product of as many factors as possible. (See Exploration 1, p. 15.)

Question 13.
60

Answer: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60

The factors of 60 are:
1 × 60 = 60
2 × 30 = 60
3 × 20 = 60
4 × 15 = 60
5 × 12 = 60
6 × 10 = 60
10 × 6 = 60
12 × 5 = 60
15 × 4 = 60
20 × 3 = 60
30 × 2 = 60
60 × 1 = 60

Question 14.
63

Answer: The factors of 63 are 1, 3, 7, 9, 21, 63

The factors of 63 are:
1 × 63 = 63
3 × 21 = 63
7 × 9 = 63
9 × 7 = 63
21 × 3 = 63
63 × 1 = 63

Question 15.
120

Answer: The factors of 120 are 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120

The factors of 120 are:
1 × 120 = 120
2 × 60 = 120
3 × 40 = 120
4 × 30 = 120
5 × 24 = 120
6 × 20 = 120
8 × 15 = 120
10 × 12 = 120
12 × 10 = 120
15 × 8 = 120
20 × 6 = 120
24 × 5 = 120
30 × 4 = 120
40 × 3 = 120
60 × 2 = 120
120 × 1 = 120

Question 16.
150

Answer: The factors of 150 are 1, 2, 3, 5, 6, 10, 15, 25, 30, 50, 75, 150

The factors of 150 are:
1 × 150 = 150
2 × 75 = 150
3 × 50 = 150
5 × 30 = 150
6 × 25 = 150
10 × 15 = 150

FINDING FACTOR PAIRS
List the factor pairs of the number.

Question 17.
15

Answer: The factor pairs of 15 are (1, 15), (3, 5)
1 × 15 = 15
3 × 5 = 15
5 × 3 = 15
15 × 1 = 15

Question 18.
22

Answer: The factor pairs of 22 are (1, 22) (2, 11)
1 × 22 = 22
2 × 11 = 22
11 × 2 = 22
22 × 1 = 22

Question 19.
34

Answer: (1, 34) (2,17)

The factor pairs of 34 are
1 × 34 = 34
2 × 17 = 34
17 × 2 = 34
34 × 1 = 34

Question 20.
39

Answer: (1, 39) (3, 13)

The factor pairs of 39 are
1 × 39 = 39
3 × 13 = 39
13 × 3 = 39
39 × 1 = 39

Question 21.
45

Answer: (1, 45) (3, 15) (5, 9)

The factor pairs of 45 are
1 × 45 = 45
3 × 15 = 45
5 × 9 = 45

Question 22.
54

Answer: (1, 54) (2, 27) (3, 18) (6, 9)

The factor pairs of 54 are
1 × 54 = 54
2 × 27 = 54
3 × 18 = 54
6 × 9 = 54

Question 23.
59

Answer: (1, 59)
The factor pairs of 59 are
1 × 59 = 59
59 × 1 = 59

Question 24.
61

Answer: (1, 61)
The factor pairs of 61 are
1 × 61 = 61
61 × 1 = 61

Question 25.
100

Answer: (1, 100) (2, 50) (4, 25) (5, 20) (10, 10)
The factor pairs of 100 are
1 × 100 = 100
2 × 50 = 100
4 × 25 = 100
5 × 20 = 100
10 × 10 = 100

Question 26.
58

Answer: (1, 58) (2, 29)
The factor pairs of 58 are
1 × 58 = 58
2 × 29 = 58

Question 27.
25

Answer: (1, 25) (5, 5)
The factor pairs of 25 are
1 × 25 = 25
5 × 5 = 25

Question 28.
76

Answer: (1, 76) (2, 38) (4, 19)
The factor pairs of 76 are
1 × 76 = 76
2 × 38 = 76
4 × 19 = 76

Question 29.
52

Answer: (1, 52) (2, 26) (4, 13)
The factor pairs of 52 are
1 × 52 = 52
2 × 26 = 52
4 × 13 = 52

Question 30.
88

Answer: (1,88) (2,44) (4, 22) (8, 11)
The factor pairs of 88 are
1 × 88 = 88
2 × 44 = 88
4 × 22 = 88
8 × 11 = 88

Question 31.
71

Answer: (1,71)
The factor pairs of 71 are
1 × 71 = 71

Question 32.
91

Answer: (1, 91) (7, 13)
The factor pairs of 91 are
1 × 91 = 91
7 × 13 = 91

WRITING A PRIME FACTORIZATION
Write the prime factorization of the number.

Question 33.
16

Answer:
16 = 2 × 8
2 × 2 × 4
2 × 2 × 2 × 2

Question 34.
25

Answer:
25 = 5 × 5

Question 35.
30

Answer:
30 = 2 × 15
= 2 × 3 × 5

Question 36.
26

Answer:
26 = 2 × 13

Question 37.
84

Answer:
84 = 2 × 42
2 × 2 × 21
2 × 2 × 3 × 7

Question 38.
54

Answer:
54 = 2 × 27
2 × 3 × 9
2 × 3 × 3 × 3

Question 39.
65

Answer:
65 = 5 × 13

Question 40.
77

Answer:
77 = 7 × 11

Question 41.
46

Answer:
46 = 2 × 23

Question 42.
39

Answer:
39 = 3 × 13

Question 43.
99

Answer:
99 = 3 × 33
3 × 3 × 11

Question 44.
24

Answer:
24 = 2 × 12
2 × 2 × 6
2 × 2 × 2 × 3

Question 45.
315

Answer:
315 = 3 × 105
3 × 3 × 35
3 × 3 × 5 × 7

Question 46.
490

Answer:
490 = 2 × 245
2 × 5 × 49
2 × 5 × 7 × 7

Question 47.
140

Answer:
2 × 70
2 × 2 × 35
2 × 2 × 5 × 7

Question 48.
640

Answer:
640 = 2 × 320
2 × 2 × 160
2 × 2 × 2 × 80
2 × 2 × 2 × 2 × 40
2 × 2 × 2 × 2 × 2 × 20
2 × 2 × 2 × 2 × 2 × 2 × 10
2 × 2 × 2 × 2 × 2 × 2 × 2 × 5

USING A PRIME FACTORIZATION
Find the number represented by the prime factorization.

Question 49.
2².3².5

Answer:
4 × 9 × 5 = 180
We have to find the prime factorization for 180.
180 = 2 × 90
2 × 2 × 45
2 × 2× 3 × 15
2 × 2 × 3 × 3 × 5

Question 50.
3².5².7

Answer:
9 × 25 × 7 = 1575
We have to find the prime factorization for 1575.
1575 = 3 × 525
3 × 3 × 175
3 × 3 × 5 × 35
3 × 3 × 5 × 5 × 7

Question 51.
2³.11².13

Answer:
8 × 11 × 13 = 1144
We have to find the prime factorization for 1144.
1144 = 2 × 572
2 × 2 × 286
2 × 2 × 2 × 143
2 × 2 × 2 × 11 × 13

Question 52.
YOU BE THE TEACHER
Your friend finds the prime factorization of 72. Is your friend correct? Explain your reasoning.

Answer:
72 = 2 × 36
2 × 2 × 18
2 × 2 × 2 × 9
2 × 2 × 2 × 3 × 3
Your friend is incorrect because you have to write the prime factorization for 9 also.
Thus the prime factorization for 72 is 2 × 2 × 2 × 3 × 3

USING A PRIME FACTORIZATION
Find the greatest perfect square that is a factor of the number.

Question 53.
250

Answer:
A = Calculate the square root of the greatest perfect square from the list of all factors of 250. The factors of 250 are 1, 2, 5, 10, 25, 50, 125, and 250. Furthermore, the greatest perfect square on this list is 25 and the square root of 25 is 5. Therefore, A equals 5.
B = Calculate 250 divided by the greatest perfect square from the list of all factors of 250. We determined above that the greatest perfect square from the list of all factors of 250 is 25. Furthermore, 250 divided by 25 is 10, therefore B equals 10.
Now we have A and B and can get our answer to 250 in its simplest radical form as follows:
√250 = A√B
√250 = 5√10

Question 54.
275

Answer:
A = Calculate the square root of the greatest perfect square from the list of all factors of 275. The factors of 275 are 1, 5, 11, 25, 55, and 275. Furthermore, the greatest perfect square on this list is 25 and the square root of 25 is 5. Therefore, A equals 5.
B = Calculate 275 divided by the greatest perfect square from the list of all factors of 275. We determined above that the greatest perfect square from the list of all factors of 275 is 25. Furthermore, 275 divided by 25 is 11, therefore B equals 11.
Now we have A and B and can get our answer to 275 in its simplest radical form as follows:
√275 = A√B
√275 = 5√11

Question 55.
392

Answer:
A = Calculate the square root of the greatest perfect square from the list of all factors of 392. The factors of 392 are 1, 2, 4, 7, 8, 14, 28, 49, 56, 98, 196, and 392. Furthermore, the greatest perfect square on this list is 196 and the square root of 196 is 14. Therefore, A equals 14.
B = Calculate 392 divided by the greatest perfect square from the list of all factors of 392. We determined above that the greatest perfect square from the list of all factors of 392 is 196. Furthermore, 392 divided by 196 is 2, therefore B equals 2.
Now we have A and B and can get our answer to 392 in its simplest radical form as follows:
√392 = A√B
√392 = 14√2

Question 56.
338

Answer:
A = Calculate the square root of the greatest perfect square from the list of all factors of 338. The factors of 338 are 1, 2, 13, 26, 169, 338. Furthermore, the greatest perfect square on this list is 324 and the square root of 324 is 18. Therefore, A equals 18.
B = Calculate 338 divided by the greatest perfect square from the list of all factors of 338. We determined above that the greatest perfect square from the list of all factors of 338 is 324.
Now we have A and B and can get our answer to 338 in its simplest radical form as follows:
√338= A√B
√392 = 18√14

Question 57.
244

Answer:
Our first step would be to find out all the factors of 244
Since this is an even number, we divide 2 we get the factors as 244/2 = 122
Next, we again divide by 2 we get the factor as 122/2 = 61
We cannot go down any further since 61 is a prime number
Since we divided by 2, 2 is itself a factor.
Lastly, we divided by 2, twice; hence 2*2 = 4 is also a factor
The factors of 244 are 2,4,61 and 122
Out of 2,4,61 and 122, the only perfect square is 4
So, the greatest perfect square that is a factor of the number 244 should be 4
Therefore, the answer is: 4

Question 58.
650

Answer:
factor 650 and find the pairs
650=2×5×5×13
the pair is 5×5 which is 5² or 25
the greatest perfect square that is a factor of 650 is 25.

Question 59.
756

Answer:
A = Calculate the square root of the greatest perfect square from the list of all factors of 756. The factors of 756 are 1, 2, 3, 4, 6, 7, 9, 12, 14, 18, 21, 27, 28, 36, 42, 54, 63, 84, 108, 126, 189, 252, 378, and 756. Furthermore, the greatest perfect square on this list is 36 and the square root of 36 is 6. Therefore, A equals 6.
B = Calculate 756 divided by the greatest perfect square from the list of all factors of 756. We determined above that the greatest perfect square from the list of all factors of 756 is 36. Furthermore, 756 divided by 36 is 21, therefore B equals 21.
Now we have A and B and can get our answer to 756 in its simplest radical form as follows:
√756 = A√B
√756 = 6√21

Question 60.
1290

Answer:
There is no greatest perfect square that is a factor of 1290.
The factors of 1290 are 1, 2, 3, 5, 6, 10, 15, 30, 43, 86, 129, 215, 258, 430, 645, 1290. There are no perfect squares as factors.

Question 61.
2205

Answer: No, the number 2,205 is not a perfect square.
The factors of 2205 are 1, 3, 5, 7, 9, 15, 21, 35, 45, 49, 63, 105, 147, 245, 315, 441, 735. There are no perfect squares as factors.

Question 62.
1890

Answer: 1890 is not the perfect square.
The factors of 1890 are 1, 3, 9, 27, 67, 201, 603, 1809. There are no perfect squares as factors.

Question 63.
495

Answer: 495 is not the perfect square.
The factors 495 are 1, 3, 5, 9, 11, 15, 33, 45, 55, 99, 165, 495. There are no perfect squares as factors.

Question 64.
4725

Answer: 4725 is not the perfect square.
The factors of 4725 are 1, 3, 5, 7, 9, 15, 21, 25, 27, 35, 45, 63, 75, 105, 135, 175, 189, 225, 315, 525, 675, 945, 1575, 4725. There are no perfect squares as factors.

Question 65.
VOCABULARY
A botanist separates plants into equal groups of 5 for an experiment. Is the total number of plants in the experiment prime or composite? Explain.

Answer: The total number of plants will be a composite.

Explanation:
She is separating them into equal groups of five. So the total number will be a multiple of 5. Five is a prime number. Multiples of prime numbers are composite numbers.

Question 66.
REASONING
A teacher divides 36 students into equal groups for a scavenger hunt. Each group should have at least 4 students but no more than 8 students. What are the possible group sizes?

Answer:
Given,
A teacher divides 36 students into equal groups for a scavenger hunt.
Each group should have at least 4 students but no more than 8 students.
The factors of 36 are 1, 2, 3, 4, 6, 9
So, there are 6 groups of 6, 9 groups of 4.

Question 67.
CRITICAL THINKING
Is 2 the only even prime number? Explain.

Answer:
The definition of a prime number is a positive integer that has exactly two distinct divisors. Since the divisors of 2 are 1 and 2, there are exactly two distinct divisors, so 2 is prime.

Question 68.
LOGIC
One table at a bake sale has 75 cookies. Another table has 60 cupcakes. Which table allows for more rectangular arrangements? Explain.

Answer:
Given,
One table at a bake sale has 75 cookies. Another table has 60 cupcakes.
75 = 3·5², so has 6 divisors. 6 rectangles are possible if you make the distinction between 1×75 and 75×1.
60 = 2²·3·5, so has 12 divisors. 12 rectangles are possible under the same conditions.

Question 69.
PERFECT NUMBERS
A perfect number is a number that equals the sum of its factors, not including itself. For example, the factors of 28 are 1, 2, 4, 7, 14, and 28. Because 1 + 2 + 4 + 7 + 14 = 28, 28 is a perfect number. What are the perfect numbers between 1 and 27?

Answer: Perfect number, a positive integer that is equal to the sum of its proper divisors. The smallest perfect number is 6, which is the sum of 1, 2, and 3. Other perfect numbers are 28, 496, and 8,128.

Question 70.
REPEATED REASONING
Choose any two perfect squares and find their product. Then multiply your answer by another perfect square. Continue this process. Are any of the products perfect squares? What can you conclude?

Answer:
2² × 3² = 4×9=36, which is a square number

Question 71.
PROBLEM SOLVING
The stage manager of a school play creates a rectangular stage that has whole number dimensions and an area of 42 square yards. String lights will outline the stage. What is the least number of yards of string lights needed to enclose the stage?

Answer:
Given that the stage manager of a school play creates a rectangular acting area of 42 square yards.
Let the length of the rectangular acting area be x, then the width is given by 42 / x.
The number of yards of string lights that the manager needs to enclose the area is given by the perimeter of the rectangular area.
Recall that the perimeter of a rectangle is given by
P = 2(length + width) = 2(x + 42/x) = 2x + 84/x
The perimeter is minimum when the differentiation of 2x + 84/x is equal to 0.
Therefore, the minimum number of yards of string lights the manager need to enclose in this area is given by
2x – 84/x = 0
2x² – 84 = 0
2x² = 84
x² = 84/2
x² = 42
x ≈ 6.48

Question 72.
DIG DEEPER!
Consider the rectangular prism shown. Using only whole number dimensions, how many different prisms are possible? Explain.

Answer:
The volume of the rectangular prism is lbh
v = 40 cubic inches
Let the length be 5 inches
breadth be 2 inches
height be 4 inches
V = 5 × 2 × 4
V = 40 cu. inches

Lesson 1.4 Greatest Common Factor

A Venn diagram uses circles to describe relationships between two or more sets. The Venn diagram shows the factors of 12 and 15. Numbers that are factors of both 12 and 15 are represented by the overlap of the two circles.

Answer: 1 and 3 are overlapped between the two circles.
So, 1 and 3 are the greatest common factors of 12 and 15.
The factors of 12 are 1, 2, 3, 4, 6, 12
The factors of 15 are 1, 3, 5, 15.

EXPLORATION 1

Identifying Common Factors

Work with a partner. In parts (a) – (d), create a Venn diagram that represents the factors of each number and identify any common factors.
a. 36 and 48
b. 16 and 56
c. 30 and 75
d. 54 and 90
e. Look at the Venn diagrams in parts (a)–(d). Explain how to identify the greatest common factor of each pair of numbers. Then circle it in each diagram.

Answer:
a.

b.

c.

d.

e. 36 and 48 have the greatest common factors.

EXPLORATION 2

Using Prime Factors
Work with a partner
a. Each Venn diagram represents the prime factorizations of two numbers. Identify each pair of numbers. Explain your reasoning.

Answer:
i. Red 2 × 3 × 3 = 18
Green 3 × 3 × 3 = 27
GCF = 9
ii. Yellow – 2 × 2 × 3 × 3 × 5 = 180
Purple – 5 × 11 = 55
GCF = 5
b. Create a Venn diagram that represents the prime factorizations of 36 and 48.

Answer:

c. Repeat part(b) for the remaining number pairs in Exploration 1.

Answer:

d. STRUCTURE
Make a conjecture about the relationship between the greatest common factors you found in Exploration 1 and the numbers in the overlaps of the Venn diagrams you just created.

Answer:
a.
The GCF between the two numbers 36 and 48 are 1,2,3,4,6,12

b.
The GCF between the two numbers 16 and 56 are 1,2,4,8
c.
The GCF between the two numbers 30 and 75 is 15.
d.

1.4 Lesson

Try It

Find the GCF of the numbers using lists of factors.

Question 1.
8, 36

Answer:
The factors of 8 are: 1, 2, 4, 8
The factors of 36 are: 1, 2, 3, 4, 6, 9, 12, 18, 36
Then the greatest common factor is 4.

Question 2.
18, 72

Answer:
The factors of 18 are: 1, 2, 3, 6, 9, 18
The factors of 72 are: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
Then the greatest common factor is 18.

Question 3.
14, 28, 49

Answer:
The factors of 14 are: 1, 2, 7, 14
The factors of 28 are: 1, 2, 4, 7, 14, 28
The factors of 49 are: 1, 7, 49
Then the greatest common factor is 7.

Another way to find the GCF of two or more numbers is by using prime factors. The GCF is the product of the common prime factors of the numbers.

Try It
Find the GCF of the numbers using prime factorizations.

Question 4.
20,45

Answer:
Find the prime factorization of 20
20 = 2 × 2 × 5
Find the prime factorization of 45
45 = 3 × 3 × 5
To find the GCF, multiply all the prime factors common to both numbers:
Therefore, GCF = 5

Question 5.
32,90

Answer:
Find the prime factorization of 32
32 = 2 × 2 × 2 × 2 × 2
Find the prime factorization of 90
90 = 2 × 3 × 3 × 5
To find the GCF, multiply all the prime factors common to both numbers:
Therefore, GCF = 2

Question 6.
45,75,120

Answer:
45= 1,3,5,9,15,45
75= 1,3,5,15,25,75
120=1,2,3,4,5,6,8,10,12,15,20,24,30,40,60,120
GCF is 15

Try It

Question 7.
Write a pair of numbers whose greatest common factor is 10.

Answer:
Let’s first find the greatest common factor (GCF) of two whole numbers. The GCF of two numbers is the greatest number that is a factor of both of the numbers. Take the numbers 50 and 30.
50 = 10 × 5
30 = 10 × 3
Their greatest common factor is 10. since 10 is the greatest factor that both numbers have in common.

Self-Assessment for Concepts & Skills

Solve each exercise. Then rate your understanding of the success criteria in your journal.

FINDING THE GCF
Find the GCF of the numbers.

Question 8.
16, 40

Answer:
The factors of 16 are: 1, 2, 4, 8, 16
The factors of 40 are: 1, 2, 4, 5, 8, 10, 20, 40
Then the greatest common factor is 8.

Question 9.
35, 63

Answer:
The factors of 35 are: 1, 5, 7, 35
The factors of 63 are: 1, 3, 7, 9, 21, 63
Then the greatest common factor is 7.

Question 10.
18, 72, 144

Answer:
The factors of 18 are: 1, 2, 3, 6, 9, 18
The factors of 72 are: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
The factors of 144 are: 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 144
Then the greatest common factor is 18.

Question 11.
MULTIPLE CHOICE
Which number is not a factor of 10? Explain.
A. 1
B. 2
C. 4
D. 5

Answer: 4

Explanation:
Factors of 10 are: 1, 2, 5, 10
Thus the correct answer is option C.

Question 12.
DIFFERENT WORDS, SAME QUESTION
Which is different? Find “both” answers.

Answer:
The Greatest Common Factor of 24 and 32 is 8
The Greatest Common Divisor of 24 and 32 is 4
The Greatest Common Prime Factor of 24 and 32 is 8
The product of common prime factors of 24 and 32 is 8.
The Greatest Common Divisor of 24 and 32 are different from others.

Self – Assessment for Problem Solving

Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 13.
You use 30 sandwiches and 42 granola bars to make identical picnic baskets. You make the greatest number of picnic baskets with no food left over. How many sandwiches and how many granola bars are in each basket?

Answer: 5 sandwiches and 7 granola in each basket

Explanation:
Given
Represent Sandwiches with S and Granola with G
S = 30
G = 42
To do this, we simply need to determine the ratio of S to G
S:G = 30:42
S:G = 5:7

Question 14.
You fill bags with cookies to give to your friends. You bake 45 chocolate chip cookies, 30 peanut butter cookies, and 15 oatmeal cookies. You want identical groups of cookies in each bag with no cookies left over. What is the greatest number of bags you can make?

Answer: 15 bags

Explanation:
Given,
You fill bags with cookies to give to your friends. You bake 45 chocolate chip cookies, 30 peanut butter cookies, and 15 oatmeal cookies.
You want identical groups of cookies in each bag with no cookies leftover.
45 chocolate chip cookies
30 peanut butter cookies
15 oatmeal cookies
So, the GCF is 15.

Greatest Common Factor Practice 1.4

Review & Refresh

List the factor pairs of the number.

Question 1.
20

Answer: The factor pairs of 20 are (1, 20) (4,5) (2,10)

Explanation:
1 × 20 = 20
4 × 5 = 20
2 × 10 = 20

Question 2.
16

Answer: The factor pairs of 16 are (1,16), (2, 8) (4,4)

Explanation:
1 × 16 = 16
2 × 8 = 16
4 × 4 = 16

Question 3.
56

Answer: The factor pairs of 56 are (1,56) (7,8) (28,2) (14,4)

Explanation:
1 × 56 = 56
7 × 8 = 56
28 × 2 = 56
14 × 4 = 56

Question 4.
87

Answer: The factor pairs of 87 are (1,87) (3,29)

Explanation:
1 × 87 = 87
3 × 29 = 87

Tell whether the statement is always, sometimes, or never true.

Question 5.
A rectangle is a rhombus.

Answer: sometimes

No, because all four sides of a rectangle don’t have to be equal. However, the sets of rectangles and rhombuses do intersect, and their intersection is the set of square all squares are both a rectangle and a rhombus.

Question 6.
A rhombus is a square.

Answer: true

A square is a special case of a rhombus because it has four equal-length sides and goes above and beyond that to also have four right angles. Every square you see will be a rhombus, but not every rhombus you meet will be a square.

Question 7.
A square is a rectangle.

Answer: not always

A square also fits the definition of a rectangle.

Question 8.
A trapezoid is a parallelogram.

Answer: never true

A trapezoid has one pair of parallel sides and a parallelogram has two pairs of parallel sides. So a parallelogram is also a trapezoid.

Concepts, Skills, & Problem Solving

USING A VENN DIAGRAM
Use a Venn diagram to find the greatest common factor of the numbers. (See Exploration 1, p. 21.)

Question 9.
12,30

Answer: 6

Question 10.
32,54

Answer: 2

Question 11.
24,108

Answer: 12

FINDING THE GCF
Find the GCF of the numbers using lists of factors.

Question 12.
6, 15

Answer: GCF is 3

Explanation:
The factors of 6 are: 1,2,3,6
The factors of 15 are: 1,3,5,15
The common Factors in 6 and 15 is 3.
Thus the greatest common factor is 3.

Question 13.
14, 84

Answer: GCF is 14

Explanation:
The factors of 14 are: 1,2,7,14
The factors of 84 are: 1,2,3,4,6,7,12,14,21,28,42 84
The greatest common factor is 14.

Question 14.
45, 108

Answer: GCF is 9

Explanation:
The factors of 45 are: 1,3,5,9,15,45
The factors of 108 are: 1,2,3,4,6,9,12,18,27,36,54,108
The greatest common factor is 9.

Question 15.
39, 65

Answer: GCF is 13

Explanation:
The factors of 39 are: 1,3,13,39
The factors of 65 are: 1,5,13,65
Thus the greatest common factor is 13.

Question 16.
51, 85

Answer: GCF is 17

Explanation:
The factors of 51 are: 1,3,17,51
The factors of 1,5,17,85
Thus the greatest common factor is 17

Question 17.
40, 63

Answer: GCF is 1

Explanation:
The factors of 40 are: 1,2,4,5,8,10,20,40
The factors of 63 are: 1,3,7,9,21,63
Thus the greatest common factor is 1.

Question 18.
12, 48

Answer: GCF is 12

Explanation:
The factors of 12 are: 1,2,3,4,6,12
The factors of 48 are: 1,2,3,4,6,8,12,16,24,48
Thus the greatest common factor is 12.

Question 19.
24, 52

Answer: GCF is 4

Explanation:
The factors of 24 are: 1,2,3,4,6,8,12,24
The factors of 1,2,4,13,36,52
Thus the greatest common factor is 4.

Question 20.
30, 58

Answer: GCF is 2

Explanation:
The factors of 30 are: 1,2,3,5,6,10,15,30
The factors of 58 are: 1,2,29,58
Thus the greatest common factor is 2.

FINDING THE GCF
Find the GCF of the numbers using prime factorizations.

Question 21.
45, 60

Answer:
The prime factorization of 45 is 3 x 3 x 5
The prime factorization 60 is 2 x 2 x 3 x 5
GCF of 45, 60 is 3 × 5 = 15

Question 22.
27, 63

Answer: 9
The prime factorization of 27 is 3 x 3 x 3
The prime factorization of 63 is 3 x 3 x 7
Thus GCF of 27, 63 is 9

Question 23.
36, 81

Answer: 9
The prime factorization of 36 is 2 x 2 x 3 x 3
The prime factorization of 81 is 3 x 3 x 3 x 3
Thus the GCF of 36, 81 is 9.

Question 24.
72, 84

Answer: 12
The prime factorization of 72 is 2 x 2 x 2 x 3 x 3
The prime factorization of 84 is 2 x 2 x 3 x 7
Thus the GCF of 72, 84 is 12.

Question 25.
61, 73

Answer: 1
The prime factorization of 61 is 1 × 61
The prime factorization 73 is 1 × 73
Thus the GCF of 61, 73 is 1

Question 26.
38, 95

Answer: 19
The prime factorization of 38 is 2 x 19
The prime factorization of 95 is 5 x 19
Thus the GCF of 38, 95 is 19

Question 27.
60, 75

Answer: 15
The prime factorization of 60 is 2 x 2 x 3 x 5
The prime factorization of 75 is 3 x 5 x 5
Thus the GCF of 60, 75 is 15.

Question 28.
42, 60

Answer: 6
The prime factorization 42 is 2 × 3 × 7
The prime factorization 60 is 2 × 2 × 3 × 5
Thus the GCF of 42, 60 is 6

Question 29.
42, 63

Answer: 21
The prime factorization of 42 is 2 × 3 × 7
The prime factorization of 63 is 3 × 3 ×7 
Thus the GCF of 42, 63 is 21

Question 30.
24, 96

Answer: 24
The prime factorization of 24 is 2 × 2 × 2 × 3
The prime factorization of 96 is 2 x 2 x 2 x 2 x 2 x 3
Thus the GCF of 24, 96 is 24.

Question 31.
189, 200

Answer: 24
The prime factorization of 189 is 3 x 3 x 3 x 7
The prime factorization of 200 is 2 x 2 x 2 x 5 x 5
Thus the GCF of 189, 200 is 24.

Question 32.
90, 108

Answer: 18
The prime factorization of 90 is 2 x 3 x 3 x 5
The prime factorization of 108 is 2 x 2 x 3 x 3 x 3
Thus the GCF of 90, 108 is 18.

OPEN-ENDED
Write a pair of numbers with the indicated GCF.

Question 33.
5

Answer: 10, 15
The factors of 10 are: 1,2,5,10
The factors of 15 are: 1,3,5,15
Thus 10, 15 are pairs of numbers with the indicated GCF 5.

Question 34.
12

Answer: 72, 84
The factors of 72 are: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
The factors of 84 are: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84
Then the greatest common factor is 12.

Question 35.
37

Answer: 37,74
The factors of 37 are: 1,37
The factors of 74 are: 1,2,37,74
Thus 37,74 are pairs of numbers with the indicated GCF 37.

Question 36.
MODELING REAL LIFE
A teacher is making identical activity packets using 92 crayons and 23 sheets of paper. What is the greatest number of packets the teacher can make with no items left over?

Answer: 23 packets

Explanation:
Factor both numbers:
1. 92=2·46=2·2·23;
2. 23 is a prime number.
Then the greatest common factor GCF (92, 23)=23.
This means that teacher can make 23 packets, each of them will contain 4 crayons and 1 sheet.

Question 37.
MODELING REAL LIFE
You are making balloon arrangements for a birthday party. There are 16 white balloons and 24 red balloons. Each arrangement must be identical. What is the greatest number of arrangements you can make using every balloon?

Answer:
This is a GCF problem. To find the GCF, list the factors.
24
1×24, 2×12, 4×6, 8×3 …
16
1×16, 2×8, 4×4, …
The greatest common factor is 8 because it is the greatest factor of both numbers.
so 8 arrangements, 2 white balloons and 3 red balloons each.
Because- 8 is the number of groups, and 8×3 =24 so 3 reds, and 2 whites because 8×2=16

YOU BE THE TEACHER
Your friend finds the GCF of the two numbers. Is your friend correct? Explain your reasoning.

Question 38.

Answer:
No, your friend is incorrect.
42 = 2 × 3 × 7
154 = 2 × 7 × 11
Thus the GCF is 14.

Question 39.

Answer: Yes your friend is correct.
Thus the GCF of 36 and 60 is 12.

FINDING THE GCF
Find the GCF of the numbers.

Question 40.
35, 56, 63

Answer: GCF is 7

Explanation:
The factors of 35 are: 1,5,7,35
The factors of 56 are: 1,2,4,7,8,14,28,56
The factors of 63 are: 1,3,7,9,21,63
Thus the greatest common factor is 7.

Question 41.
30, 60, 78

Answer: GCF is 6

Explanation:
The factors of 30 are: 1,2,3,5,6,10,15,30
The factors of 60 are: 1,2,3,4,5,6,10,12,15,20,30,60
The factors of 78 are: 1,2,3,6,13,26,39,78
Thus the greatest common factor is 6.

Question 42.
42, 70, 84

Answer: GCF is 14

The factors of 42 are: 1, 2, 3, 6, 7, 14, 21, 42
The factors of 70 are: 1, 2, 5, 7, 10, 14, 35, 70
The factors of 84 are: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84
Then the greatest common factor is 14.

Question 43.
40, 55, 72

Answer: GCF is 1

The factors of 40 are: 1, 2, 4, 5, 8, 10, 20, 40
The factors of 55 are: 1, 5, 11, 55
The factors of 72 are: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
Then the greatest common factor is 1.

Question 44.
18, 54, 90

Answer: GCF is 18

The factors of 18 are: 1, 2, 3, 6, 9, 18
The factors of 54 are: 1, 2, 3, 6, 9, 18, 27, 54
The factors of 90 are: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90
Then the greatest common factor is 18.

Question 45.
16, 48, 88

Answer: GCF is 8

The factors of 16 are: 1, 2, 4, 8, 16
The factors of 48 are: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48
The factors of 88 are: 1, 2, 4, 8, 11, 22, 44, 88
Then the greatest common factor is 8.

Question 46.
52, 78, 104

Answer: GCF is 26

The factors of 52 are: 1, 2, 4, 13, 26, 52
The factors of 78 are: 1, 2, 3, 6, 13, 26, 39, 78
The factors of 104 are: 1, 2, 4, 8, 13, 26, 52, 104
Then the greatest common factor is 26.

Question 47.
96, 120, 156

Answer: GCF is 12

The factors of 96 are: 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96
The factors of 120 are: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120
The factors of 156 are: 1, 2, 3, 4, 6, 12, 13, 26, 39, 52, 78, 156
Then the greatest common factor is 12.

Question 48.
280, 300, 380

Answer: GCF is 20

The factors of 280 are: 1, 2, 4, 5, 7, 8, 10, 14, 20, 28, 35, 40, 56, 70, 140, 280
The factors of 300 are: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 25, 30, 50, 60, 75, 100, 150, 300
The factors of 380 are: 1, 2, 4, 5, 10, 19, 20, 38, 76, 95, 190, 380
Then the greatest common factor is 20.

Question 49.
OPEN-ENDED
Write three numbers that have a GCF of 16. What method did you use to find your answer?

Answer: 16, 32, 48

The factors of 16 are: 2 × 2 × 2 × 2
The factors of 32 are : 2 × 2 × 2 × 2 × 2
The factors of 48 are: 3 × 2 × 2 × 2 × 2

CRITICAL THINKING
Tell whether the statement is always, sometimes, or never true. Explain your reasoning.

Question 50.
The GCF of two even numbers is 2.

Answer: Always

Explanation:
Example:
The factors of 14 are: 1, 2, 7, 14
The factors of 16 are: 1, 2, 4, 8, 16
Then the greatest common factor is 2.

Question 51.
The GCF of two prime numbers is 1.

Answer: Always

Explanation:
Example:
The factors of 3 are: 1, 3
The factors of 5 are: 1, 5
Then the greatest common factor is 1.

Question 52.
When one number is a multiple of another, the GCF of the numbers is the greater of the numbers.

Answer:
When one number is a multiple of another, the GCF of the numbers is the greater of the numbers. This is never true since the GCF is a factor of both numbers. So the GCF is the smaller of the two numbers.

Question 53.
PROBLEM SOLVING
A science museum makes gift bags for students using 168 magnets, 48 robot figurines, and 24 packs of freeze-dried ice cream. What is the greatest number of gift bags that can be made using all of the items? How many of each item are in each gift bag?

Answer:
The greatest common factor of 24, 48, and 168 is 24, so 24 gift bags can be made. Each will have 1/24 of the number of gift items of each type that are available.
In each bag are
1/24 × 168 magnets = 7 magnets
1/24 × 48 robot figurines = 2 robot figurines
1/24 × 24 packs of ice cream = 1 pack of ice cream

Question 54.
VENN DIAGRAM
Consider the numbers 252, 270, and 300.
a. Create a Venn diagram using the prime factors of the numbers.

Answer:

b. Use the Venn diagram to find the GCF of 252, 270, and 300.

Answer:
The factors of 252 are: 1, 2, 3, 4, 6, 7, 9, 12, 14, 18, 21, 28, 36, 42, 63, 84, 126, 252
The factors of 270 are: 1, 2, 3, 5, 6, 9, 10, 15, 18, 27, 30, 45, 54, 90, 135, 270
The factors of 300 are: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 25, 30, 50, 60, 75, 100, 150, 300
Then the greatest common factor is 6.
c. What is the GCF of 252 and 270? 252 and 300? 270 and 300? Explain how you found your answers.

Answer:
The factors of 252 are: 1, 2, 3, 4, 6, 7, 9, 12, 14, 18, 21, 28, 36, 42, 63, 84, 126, 252
The factors of 270 are: 1, 2, 3, 5, 6, 9, 10, 15, 18, 27, 30, 45, 54, 90, 135, 270
The GCF of 252 and 270 is 18.
252 and 300:
The factors of 252 are: 1, 2, 3, 4, 6, 7, 9, 12, 14, 18, 21, 28, 36, 42, 63, 84, 126, 252
The factors of 300 are: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 25, 30, 50, 60, 75, 100, 150, 300
Then the greatest common factor is 12.
270 and 300:
The factors of 270 are: 1, 2, 3, 5, 6, 9, 10, 15, 18, 27, 30, 45, 54, 90, 135, 270
The factors of 300 are: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 25, 30, 50, 60, 75, 100, 150, 300
Then the greatest common factor is 30.

Question 55.
REASONING
You are making fruit baskets using 54 apples, 36 oranges, and 73 bananas.
a. Explain why you cannot make identical fruit baskets without leftover fruit.

Answer: 73 is a prime number. It can only be divided by 1 and by itself.
b. What is the greatest number of identical fruit baskets you can make with the least amount of fruit left over? Explain how you found your answer.

Answer:
The GCF of the three numbers:
54 36 73
1×54 1×36 1×73
2×27 2×18
3×18 3×12
6×9 4×9
6×6
GCF of 54, 36, and 73 is 1
GCF of 54 and 36 is 18
If we divide 54 apples into 18 baskets, we have 3 apples in each basket
If we divide 36 oranges into 18 baskets, we have 2 oranges in each basket
If we divide 73 bananas into 18 baskets, we have 4 bananas in each basket + one banana left over.
So the greatest number of identical fruit baskets we can make with the least amount of fruit left over is 18 baskets

Question 56.
DIG DEEPER!
Two rectangular, adjacent rooms share a wall. One-foot-by-one-foot tiles cover the floor of each room. Describe how the greatest possible length of the adjoining wall is related to the total number of tiles in each room. Draw a diagram that represents one possibility.

Answer:
Consider two adjacent rectangular rooms having Length=L, and, Breadth = B
Now Suppose the wall which is in between two rooms has a height or length =H.
The breadth of wall = B [ if the wall doesn’t exceed the breadth of the room]
Considering two rooms to be identical,
Area of each room= L × B square unit
Area of each tile = 1×1=1 square unit
Number of tiles required= L B ÷ 1= LB tiles( product of length and breadth of the room is the number of tiles required)
Suppose if,LB= N
B= N/LArea of the wall(W) = B×H= B H square unit
B =W/H
Equating (1) and (2)
N/L = W/ H
H = WL/N
H = WL/LB
H = W/B
H = Area of wall/Breadth of room or wall

Lesson 1.5 Least Common Multiple

EXPLORATION 1
Identifying Common Multiples

Work with a partner. In parts (a)–(d), create a Venn diagram that represents the first several multiples of each number and identify any common multiples.
a. 8 and 12
b. 4 and 14
c. 10 and 15
d. 20 and 35
e. Look at the Venn diagrams in parts (a)–(d). Explain how to identify the least common multiple of each pair of numbers. Then circle it in each diagram.

Answer:
a.
b.
c.

d.

EXPLORATION 2
Using Prime Factors
Work with a partner.
a. Create a Venn diagram that represents the prime factorizations of 8 and 12.

Answer:

b. Repeat part (a) for the remaining number pairs in Exploration 1.

Answer:

c. STRUCTURE
Make a conjecture about the relationship between the least common multiples you found in Exploration 1 and the numbers in the Venn diagrams you just created.

Answer:
The numbers which are overlapped are the least common multiples of the numbers.
d. The Venn diagram shows the prime factors of two numbers.

Use the diagram to complete the following tasks.

• Identify the two numbers.
• Find the greatest common factor.
• Find the least common multiple

Answer:

• 120, 180
• The factors of 120 are: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120The factors of 180 are: 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, 180Then the greatest common factor is 60.
• Find and list multiples of each number until the first common multiple is found. This is the lowest common multiple.
  Multiples of 120:
  120, 240, 360, 480, 600
  Multiples of 180:
  180, 360, 540, 720
  Therefore,
  LCM(120, 180) = 360

1.5 Lesson

Try It

Find the LCM of the numbers using lists of multiples.

Question 1.
3, 8

Answer: 24
Find and list multiples of each number until the first common multiple is found. This is the lowest common multiple.
Multiples of 3:
3, 6, 9, 12, 15, 18, 21, 24, 27, 30
Multiples of 8:
8, 16, 24, 32, 40
Therefore,
LCM(3, 8) = 24

Question 2.
9, 12

Answer:
Find and list multiples of each number until the first common multiple is found. This is the lowest common multiple.
Multiples of 9:
9, 18, 27, 36, 45, 54
Multiples of 12:
12, 24, 36, 48, 60
Therefore,
LCM(9, 12) = 36

Question 3.
6, 10

Answer:
Find and list multiples of each number until the first common multiple is found. This is the lowest common multiple.
Multiples of 6:
6, 12, 18, 24, 30, 36, 42
Multiples of 10:
10, 20, 30, 40, 50
Therefore,
LCM(6, 10) = 30

Try It
Find the LCM of the numbers using prime factorizations.

Question 4.
14, 18

Answer:
List all prime factors for each number.
Prime Factorization of 14 is:
2 x 7
Prime Factorization of 18 is:
2 x 3 x 3
For each prime factor, find where it occurs most often as a factor and write it that many times in a new list.
The new superset list is
2, 3, 3, 7
Multiply these factors together to find the LCM.
LCM = 2 x 3 x 3 x 7 = 126
LCM(14, 18) = 126

Question 5.
28, 36

Answer:
List all prime factors for each number.
Prime Factorization of 28 is:
2 x 2 x 7
Prime Factorization of 36 is:
2 x 2 x 3 x 3
For each prime factor, find where it occurs most often as a factor and write it that many times in a new list.
The new superset list is
2, 2, 3, 3, 7
Multiply these factors together to find the LCM.
LCM = 2 x 2 x 3 x 3 x 7 = 252
LCM = 252
Therefore,
LCM(28, 36) = 252

Question 6.
24, 90

Answer:
List all prime factors for each number.
Prime Factorization of 24 is:
2 x 2 x 2 x 3
Prime Factorization of 90 is:
2 x 3 x 3 x 5
For each prime factor, find where it occurs most often as a factor and write it that many times in a new list.
The new superset list is
2, 2, 2, 3, 3, 5
Multiply these factors together to find the LCM.
LCM = 2 x 2 x 2 x 3 x 3 x 5 = 360
LCM = 360
Therefore,
LCM(24, 90) = 360

Try It

Find the LCM of the numbers.

Question 7.
2, 5, 8

Answer:
Find and list multiples of each number until the first common multiple is found. This is the lowest common multiple.
Multiples of 2:
2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44
Multiples of 5:
5, 10, 15, 20, 25, 30, 35, 40, 45, 50
Multiples of 8:
8, 16, 24, 32, 40, 48, 56
Therefore,
LCM(2, 5, 8) = 40

Question 8.
6, 10, 12

Answer:
Find and list multiples of each number until the first common multiple is found. This is the lowest common multiple.
Multiples of 6:
6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72
Multiples of 10:
10, 20, 30, 40, 50, 60, 70, 80
Multiples of 12:
12, 24, 36, 48, 60, 72, 84
Therefore,
LCM(6, 10, 12) = 60

Question 9.
Write three numbers that have a least common multiple of 100.

Answer: 1, 10, 100

Explanation:
Find and list multiples of each number until the first common multiple is found. This is the lowest common multiple.
Multiples of 1:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102
Multiples of 10:
10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120
Multiples of 100:
100, 200, 300
Therefore,
LCM(1, 10, 100) = 100

Self-Assessment for Concepts & Skills

Solve each exercise. Then rate your understanding of the success criteria in your journal.

FINDING THE LCM
Find the LCM of the numbers.

Question 10.
6, 9

Answer:
Find and list multiples of each number until the first common multiple is found. This is the lowest common multiple.
Multiples of 6:
6, 12, 18, 24, 30
Multiples of 9:
9, 18, 27, 36
Therefore,
LCM(6, 9) = 18

Question 11.
30, 40

Answer:
Find and list multiples of each number until the first common multiple is found. This is the lowest common multiple.
Multiples of 30:
30, 60, 90, 120, 150, 180
Multiples of 40:
40, 80, 120, 160, 200
Therefore,
LCM(30, 40) = 120

Question 12.
5, 11

Answer:
Find and list multiples of each number until the first common multiple is found. This is the lowest common multiple.
Multiples of 5:
5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65
Multiples of 11:
11, 22, 33, 44, 55, 66, 77
Therefore,
LCM(5, 11) = 55

Question 13.
Reasoning
Write two numbers such that 18 and 30 are multiples of the numbers. Justify your answer.

Answer: 3, 6

Explanation:
Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30
Multiples of 6: 6,12,18,24,30,36,42,48
Thus 18 and 30 are the multiples of 3 and 6.

Question 14.
REASONING
You need to find the LCM of 13 and 14. Would you rather list their multiples or use their prime factorizations? Explain.

Answer:
List all prime factors for each number.
Prime Factorization of 13 shows:
13 is prime = 13
Prime Factorization of 14 is:
2 x 7
For each prime factor, find where it occurs most often as a factor and write it that many times in a new list.
The new superset list is
2, 7, 13
Multiply these factors together to find the LCM.
LCM = 2 x 7 x 13 = 182
In exponential form:
LCM = 2 x 7 x 13 = 182
LCM = 182
Therefore,
LCM(13, 14) = 182

Question 15.
CHOOSE TOOLS
A student writes the prime factorizations of 8 and 12 in a table as shown. She claims she can use the table to find the greatest common factor and the least common multiple of 8 and 12. How is this possible?

Answer:

For each prime factor, find where it occurs most often as a factor and write it that many times in a new list.
The new superset list is
2, 2, 2, 3
Multiply these factors together to find the LCM.
LCM = 2 x 2 x 2 x 3 = 24
The least common multiple of 8 and 12 is 24.

Self-Assessment for Problem Solving

Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 17.
A geyser erupts every fourth day. Another geyser erupts every sixth day. Today both geysers erupted. In how many days will both geysers erupt on the same day again?

Answer:
Given,
A geyser erupts every fourth day. Another geyser erupts every sixth day. Today both geysers erupted.
The geyser erupts on the same day after 12 days.

Question 18.
A water park has two large buckets that slowly fill with water. One bucket dumps water every 12 minutes. The other bucket dumps water every 10 minutes. Five minutes ago, both buckets dumped water. When will both buckets dump water at the same time again?

Answer:
Given,
A water park has two large buckets that slowly fill with water.
One bucket dumps water every 12 minutes.
The other bucket dumps water every 10 minutes.
Five minutes ago, both buckets dumped water.
Both buckets will dump again at the same time in 60 minutes (1 hour.)

Question 19.
DIG DEEPER!
You purchase disposable plates, cups, and forks for a cookout. Plates are sold in packages of 24, cups in packages of 32, and forks in packages of 48. What are the least numbers of packages you should buy in order to have the same number of plates, cups, and forks?

Answer:
Given,
You purchase disposable plates, cups, and forks for a cookout. Plates are sold in packages of 24, cups in packages of 32, and forks in packages of 48.
We solve this question using the Lowest Common Multiple (LCM) method/
Step 1
We list multiples of each number until the first common multiple is found. This is referred to as the lowest common multiple.
Plates are sold in packages of 24
Cups in packages of 32
Forks in packages of 48
Multiples of 24:
24, 48, 72, 96, 120, 144
Multiples of 32:
32, 64, 96, 128, 160
Multiples of 48:
48, 96, 144, 192
Therefore,
LCM(24, 32, 48) = 96
Hence, the least numbers of packages you should buy in order to have the same number of plates, cups, and forks is 96 packages

Least Common Multiple Practice 1.5

Review & Refresh

Find the GCF of the numbers.

Question 1.
18, 42

Answer:
The factors of 18 are: 1, 2, 3, 6, 9, 18
The factors of 42 are: 1, 2, 3, 6, 7, 14, 21, 42
Then the greatest common factor is 6.

Question 2.
72, 96

Answer:
The factors of 72 are: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
The factors of 96 are: 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96
Then the greatest common factor is 24.

Question 3.
38, 76, 114

Answer:
The factors of 38 are: 1, 2, 19, 38
The factors of 76 are: 1, 2, 4, 19, 38, 76
The factors of 114 are: 1, 2, 3, 6, 19, 38, 57, 114
Then the greatest common factor is 38.

Divide.

Question 4.
900 ÷ 6

Answer: 150

Explanation:
Divide the two numbers 900 and 6.
900/6 = 150
It means 6 divides 900 150 times.
Thus the quotient is 150.

Question 5.
1944 ÷ 9

Answer: 216

Explanation:
Divide the two numbers 1944 and 9
1944/9 = 216
It means that 9 divides 1944 216 times.
Thus the quotient is 216

Question 6.
672 ÷ 12

Answer: 56

Explanation:
Divide the two numbers 672 and 12.
672/12 = 56
It means 12 divides 672 56 times.
Thus the quotient is 56.

Write an ordered pair that corresponds to the point.

Question 7.
Point A

Answer: (2,4)
By seeing the below graph we can find the ordered pairs.
2 lies on the x-axis and 4 lies on the y-axis.
So, the ordered pairs to the point A is (2,4)

Question 8.
Point B

Answer: (3,1)
By seeing the below graph we can find the ordered pairs.
3 lies on the x-axis and 1 lies on the y-axis.
So, the ordered pairs to the point A is (3,1)

Question 9.
Point C

Answer: (4,7)
By seeing the below graph we can find the ordered pairs.
4 lies on the x-axis and 7 lies on the y-axis.
So, the ordered pairs to the point A is (4,7)

Question 10.
Point D

Answer: (9,6)
By seeing the below graph we can find the ordered pairs.
9 lies on the x-axis and 6 lies on the y-axis.
So, the ordered pairs to the point A is (9,6)

Concepts, Skills, & Problem Solving

USING A VENN DIAGRAM
Use a Venn diagram to find the least common multiple of the numbers. (See Exploration 1, p. 27.)

Question 11.
3, 7

Answer: 21

Question 12.
6, 8

Answer: 24

Question 13.
4, 5

Answer:

FINDING THE LCM
Find the LCM of the numbers using lists of multiples.

Question 14.
1, 5

Answer:
Find and list multiples of each number until the first common multiple is found. This is the lowest common multiple.
Multiples of 1:
1, 2, 3, 4, 5, 6, 7
Multiples of 5:
5, 10, 15
Therefore,
LCM(1, 5) = 5

Question 15.
2, 6

Answer:
Find and list multiples of each number until the first common multiple is found. This is the lowest common multiple.
Multiples of 2:
2, 4, 6, 8, 10
Multiples of 6:
6, 12, 18
Therefore,
LCM(2, 6) = 6

Question 16.
2, 3

Answer:
Find and list multiples of each number until the first common multiple is found. This is the lowest common multiple.
Multiples of 2:
2, 4, 6, 8, 10
Multiples of 3:
3, 6, 9, 12
Therefore,
LCM(2, 3) = 6

Question 17.
2, 9

Answer:
Find and list multiples of each number until the first common multiple is found. This is the lowest common multiple.
Multiples of 2:
2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22
Multiples of 9:
9, 18, 27, 36
Therefore,
LCM(2, 9) = 18

Question 18.
3, 4

Answer:
Find and list multiples of each number until the first common multiple is found. This is the lowest common multiple.
Multiples of 3:
3, 6, 9, 12, 15, 18
Multiples of 4:
4, 8, 12, 16, 20
Therefore,
LCM(3, 4) = 12

Question 19.
8, 9

Answer:
Find and list multiples of each number until the first common multiple is found. This is the lowest common multiple.
Multiples of 8:
8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88
Multiples of 9:
9, 18, 27, 36, 45, 54, 63, 72, 81, 90
Therefore,
LCM(8, 9) = 72

Question 20.
5, 8

Answer:
Find and list multiples of each number until the first common multiple is found. This is the lowest common multiple.
Multiples of 5:
5, 10, 15, 20, 25, 30, 35, 40, 45, 50
Multiples of 8:
8, 16, 24, 32, 40, 48, 56
Therefore,
LCM(5, 8) = 40

Question 21.
11, 12

Answer:
Find and list multiples of each number until the first common multiple is found. This is the lowest common multiple.
Multiples of 11:
11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 121, 132, 143, 154
Multiples of 12:
12, 24, 36, 48, 60, 72, 84, 96, 108, 120, 132, 144, 156
Therefore,
LCM(11, 12) = 132

Question 22.
12, 18

Answer:
Find and list multiples of each number until the first common multiple is found. This is the lowest common multiple.
Multiples of 12:
12, 24, 36, 48, 60
Multiples of 18:
18, 36, 54, 72
Therefore,
LCM(12, 18) = 36

FINDING THE LCM
Find the LCM of the numbers using prime factorizations.

Question 23.
7, 12

Answer:
List all prime factors for each number.
Prime Factorization of 7 shows:
7 is prime
Prime Factorization of 12 is:
2 x 2 x 3
For each prime factor, find where it occurs most often as a factor and write it that many times in a new list.
The new superset list is
2, 2, 3, 7
Multiply these factors together to find the LCM.
LCM = 2 x 2 x 3 x 7 = 84
LCM = 84
Therefore,
LCM(7, 12) = 84

Question 24.
5, 9 4

Answer:
List all prime factors for each number.
Prime Factorization of 4 is:
2 x 2
Prime Factorization of 5 shows:
5 is prime
Prime Factorization of 9 is:
3 x 3
For each prime factor, find where it occurs most often as a factor and write it that many times in a new list.
The new superset list is
2, 2, 3, 3, 5
Multiply these factors together to find the LCM.
LCM = 2 x 2 x 3 x 3 x 5 = 180
LCM = 180
Therefore,
LCM(4, 5, 9) = 180

Question 25.
4, 11

Answer:
List all prime factors for each number.
Prime Factorization of 4 is:
2 x 2
Prime Factorization of 11 shows:
11 is prime
For each prime factor, find where it occurs most often as a factor and write it that many times in a new list.
The new superset list is
2, 2, 11
Multiply these factors together to find the LCM.
LCM = 2 x 2 x 11 = 44
LCM = 44
Therefore,
LCM(4, 11) = 44

Question 26.
9, 10

Answer:
List all prime factors for each number.
Prime Factorization of 9 is:
3 x 3
Prime Factorization of 10 is:
2 x 5
For each prime factor, find where it occurs most often as a factor and write it that many times in a new list.
The new superset list is
2, 3, 3, 5
Multiply these factors together to find the LCM.
LCM = 2 x 3 x 3 x 5 = 90
LCM = 90
Therefore,
LCM(9, 10) = 90

Question 27.
12, 27

Answer:
List all prime factors for each number.
Prime Factorization of 12 is:
2 x 2 x 3
Prime Factorization of 27 is:
3 x 3 x 3
For each prime factor, find where it occurs most often as a factor and write it that many times in a new list.
The new superset list is
2, 2, 3, 3, 3
Multiply these factors together to find the LCM.
LCM = 2 x 2 x 3 x 3 x 3 = 108
LCM = 108
Therefore,
LCM(12, 27) = 108

Question 28.
18, 45

Answer:
List all prime factors for each number.
Prime Factorization of 18 is:
2 x 3 x 3
Prime Factorization of 45 is:
3 x 3 x 5
For each prime factor, find where it occurs most often as a factor and write it that many times in a new list.
The new superset list is
2, 3, 3, 5
Multiply these factors together to find the LCM.
LCM = 2 x 3 x 3 x 5 = 90
LCM = 90
Therefore,
LCM(18, 45) = 90

Question 29.
22, 23

Answer:
List all prime factors for each number.
Prime Factorization of 22 is:
2 x 11
Prime Factorization of 23 shows:
23 is prime
For each prime factor, find where it occurs most often as a factor and write it that many times in a new list.
The new superset list is
2, 11, 23
Multiply these factors together to find the LCM.
LCM = 2 x 11 x 23 = 506
LCM = 506
Therefore,
LCM(22, 23) = 506

Question 30.
36, 60

Answer:
List all prime factors for each number.
Prime Factorization of 36 is:
2 x 2 x 3 x 3
Prime Factorization of 60 is:
2 x 2 x 3 x 5
For each prime factor, find where it occurs most often as a factor and write it that many times in a new list.
The new superset list is
2, 2, 3, 3, 5
Multiply these factors together to find the LCM.
LCM = 2 x 2 x 3 x 3 x 5 = 180
LCM = 180
Therefore,
LCM(36, 60) = 180

Question 31.
35, 50

Answer:
List all prime factors for each number.
Prime Factorization of 35 is:
5 x 7
Prime Factorization of 50 is:
2 x 5 x 5
For each prime factor, find where it occurs most often as a factor and write it that many times in a new list.
The new superset list is
2, 5, 5, 7
Multiply these factors together to find the LCM.
LCM = 2 x 5 x 5 x 7 = 350
LCM = 350
Therefore,
LCM(35, 50) = 350

Question 32.
YOU BE THE TEACHER
Your friend finds the LCM of 6 and 9. Is your friend correct? Explain your reasoning.

Answer: No friend is incorrect.
List all prime factors for each number.
Prime Factorization of 6 is:
2 x 3
Prime Factorization of 9 is:
3 x 3
For each prime factor, find where it occurs most often as a factor and write it that many times in a new list.
The new superset list is
2, 3, 3
Multiply these factors together to find the LCM.
LCM = 2 x 3 x 3 = 18
LCM = 18
Therefore,
LCM(6, 9) = 18

Question 33.
MODELING REAL LIFE
You have diving lessons every fifth day and swimming lessons every third day. Today you have both lessons. In how many days will you have both lessons on the same day again?

Answer:
After 15 days
diving is on 0 , 5 , 10 , 15 , 20 , … days
swimming is on 0 , 3, 6 , 9, 12 , 15 , 18 , … days
Thus the next will be in 15 days

Question 34.
REASONING
Which model represents an LCM that is different from the other three? Explain your reasoning.

Answer: The number line B is different from the other three number lines.

FINDING THE LCM
Find the LCM of the numbers.

Question 35.
2, 3, 7

Answer:
LCM(2,3) = (2 × 3) / GCF(2,3)
= (2 × 3) / 1
= 6 / 1
= 6
LCM(6,7) = (6 × 7) / GCF(6,7)
= (6 × 7) / 1
= 42 / 1
= 42
Therefore,
LCM(2, 3, 7) = 42

Question 36.
3, 5, 11

Answer:
LCM(3,5) = (3 × 5) / GCF(3,5)
= (3 × 5) / 1
= 15 / 1
= 15
LCM(15,11) = (15 × 11) / GCF(15,11)
= (15 × 11) / 1
= 165 / 1
= 165
Therefore,
LCM(3, 5, 11) = 165

Question 37.
4, 9, 12

Answer:
LCM(4,9) = (4 × 9) / GCF(4,9)
= (4 × 9) / 1
= 36 / 1
= 36
LCM(36,12) = (36 × 12) / GCF(36,12)
= (36 × 12) / 12
= 432 / 12
= 36
Therefore,
LCM(4, 9, 12) = 36

Question 38.
6, 8, 15

Answer:
LCM(6,8) = (6 × 8) / GCF(6,8)
= (6 × 8) / 2
= 48 / 2
= 24
LCM(24,15) = (24 × 15) / GCF(24,15)
= (24 × 15) / 3
= 360 / 3
= 120
Therefore,
LCM(6, 8, 15) = 120

Question 39.
7, 18, 21

Answer:
LCM(7,18) = (7 × 18) / GCF(7,18)
= (7 × 18) / 1
= 126 / 1
= 126
LCM(126,21) = (126 × 21) / GCF(126,21)
= (126 × 21) / 21
= 2646 / 21
= 126
Therefore,
LCM(7, 18, 21) = 126

Question 40.
9, 10, 28

Answer:
LCM(9,10) = (9 × 10) / GCF(9,10)
= (9 × 10) / 1
= 90 / 1
= 90
LCM(90,28) = (90 × 28) / GCF(90,28)
= (90 × 28) / 2
= 2520 / 2
= 1260
Therefore,
LCM(9, 10, 28) = 1260

Question 41.
PROBLEM SOLVING
At Union Station, you notice that three subway lines just arrived at the same time. How long must you wait until all three lines arrive at Union Station at the same time again?

Answer: 60 minutes

Explanation:
The complete question in the attached figure
Step 1
Find the least common multiple (LCM) of the three numbers
List the prime factors of each number
10 = 2 × 5
12 = 2 × 2 × 3
15 = 3 × 5
Multiply each factor the greatest number of times it occurs in any of the numbers to find out the LCM
The LCM is equal to
4 × 3 × 5 = 60
Thus You must wait 60 minutes for all three lines to arrive at Union Station at the same time again.

Question 42.
DIG DEEPER!
A radio station gives away $15 to every 15th caller, $25 to every 25th caller, and a free concert ticket to every 100th caller. When will the station first give away all three prizes to one caller? When this happens, how much money and how many tickets are given away?

Answer:
Given,
Radio Station gives :
1st prize: $15 to 15th caller
2nd prize: $25 to 25th caller
3rd prize: free concert tickets to 100th caller
So, in order to get all three prizes the caller must be 15th, 25th, and 100th caller at the same time. But to find when the radio station will give first all three prizes we calculate L.C.M. of ( 15, 25, 100 ) that is 300
Hence, the station first gives away all three prizes to the 300th caller.

Question 43.
LOGIC
You and a friend are running on treadmills. You run 0.5 mile every 3 minutes, and your friend runs 2 miles every 14 minutes. You both start and stop running at the same time and run a whole number of miles. What are the least possible numbers of miles you and your friend can run?

Answer:
If you run 0.5 miles every 3 minutes then you run 1 mile every 6 minutes.
If your friend runs 2 miles every 14 minutes then your friend runs 1 mile every 7 minutes.
You will both then both run a whole number of minutes for a time that is a multiple of 6 and 7.
The least common multiple of 6 and 7 is 42 so the least possible time you and your friend could run for and both run a whole number of miles is then 42 minutes.
Since you run 1 mile every 6 minutes, in 42 minutes you will run 42/6=7 miles.
Since your friend runs 1 mile every 7 minutes, in 42 minutes your friend will run 42/7=6 miles.

Question 44.
VENN DIAGRAM

Refer to the Venn diagram.
a. Copy and complete the Venn diagram.
b. What is the LCM of 16, 24, and 40?
c. What is the LCM of 16 and 40? 24 and 40? 16 and 24? Explain how you found your answers.

Answer:

Find and list multiples of each number until the first common multiple is found. This is the lowest common multiple.
LCM of 16, 40:
Multiples of 16:
16, 32, 48, 64, 80, 96, 112
Multiples of 40:
40, 80, 120, 160
Therefore,
LCM(16, 40) = 80
LCM of 24, 40:
Multiples of 24:
24, 48, 72, 96, 120, 144, 168
Multiples of 40:
40, 80, 120, 160, 200
Therefore,
LCM(24, 40) = 120
LCM of 16, 24:
Multiples of 16:
16, 32, 48, 64, 80
Multiples of 24:
24, 48, 72, 96
Therefore,
LCM(16, 24) = 48

CRITICAL THINKING
Tell whether the statement is always, sometimes, or never true. Explain your reasoning.

Question 45.
The LCM of two different prime numbers is their product.

Answer: Always true

Example:
3 × 5 = 15

Question 46.
The LCM of a set of numbers is equal to one of the numbers in the set.

Answer:
The LCM of a set of numbers is equal to one of the numbers in the set. Always Sometimes Never true. Question 909193: The LCM of a set of numbers is equal to one of the numbers in the set. This is sometimes true.

Question 47.
The GCF of two different numbers is the LCM of the numbers.

Answer:
Another way to find the LCM of two numbers is to divide their product by their greatest common factor ( GCF ). Example 2: Find the least common multiple of 18 and 20. The common factors are 2 and 3 .

Numerical Expressions and Factors Connecting Concepts

Getting Ready for Chapter Connecting Concepts

Using the Problem-Solving Plan

Question 1.
A sports team gives away shirts at the stadium. There are 60 large shirts, 1.6 times as many small shirts as large shirts, and 1.5 times as many medium shirts as small shirts. The team wants to divide the shirts into identical groups to be distributed throughout the stadium. What is the greatest number of groups that can be formed using every shirt?

Understand the Problem
You know the number of large shirts and two relationships among the numbers of small, medium, and large shirts. You are asked to find the greatest number of identical groups that can be formed using every shirt.
Make a plan
Break the problem into parts. First use multiplication to find the number of each size shirt. Then find the GCF of these numbers.
Solve and Check
Use the plan to solve the problem. Then check your solution.

Answer:
Given,
A sports team gives away shirts at the stadium.
There are 60 large shirts, 1.6 times as many small shirts as large shirts, and 1.5 times as many medium shirts as small shirts. The team wants to divide the shirts into identical groups to be distributed throughout the stadium.
60 × 1.6 = 96
60 × 1.5 = 90
The factors of 90 are: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90
The factors of 96 are: 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96
Then the greatest common factor is 6.

Question 2.
An escape artist fills the tank shown with water. Find the number of cubic feet of water needed to fill the tank. Then find the number of cubic yards of water that are needed to fill the tank. Justify your answer.