Question 1.
\na. What is the decimal expansion of the number \\(\\frac{35}{7}\\)? Is the number \\(\\frac{35}{7}\\) rational or irrational? Explain.
\nAnswer:
\n\\(\\frac{35}{7}\\) = 5.000….
\nThe number \\(\\frac{35}{7}\\) is a rational number. Rational numbers have decimal expansions that repeat. In this case, the decimal that repeats is a zero.<\/p>\n
b. What is the decimal expansion of the number \\(\\frac{4}{33}\\)? Is the number \\(\\frac{4}{33}\\) rational or irrational? Explain.
\nAnswer:
\n\\(\\frac{4}{33}\\) = 0.1212……… = \\(0 . \\overline{12}\\)
\n![\"Engage](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/04\/Engage-NY-Math-8th-Grade-Module-7-Mid-Module-Assessment-Answer-Key-3.png\")
\nThe number \\(\\frac{4}{33}\\) is a rational number. Rational numbers have decimal expansions that repeat. The digits 12 repeat in the decimal expansion of \\(\\frac{4}{33}\\). So, \\(\\frac{4}{33}\\) is rational.<\/p>\n
Question 2. b. Write \\(2.8 \\overline{40}\\) as a fraction. Let y be \\(0. \\overline{40}\\) c. Brandon stated that 0.66 and \\(\\frac{2}{3}\\) are equivalent. Do you agree? Explain why or why not. d. Between which two positive integers does \\(\\sqrt{33}\\) lie? e. For what integer x is \\(\\sqrt{x}\\) closest to 5.25? Explain. Question 3. b. \\(5 . \\overline{39}\\) c. \\(\\frac{12}{4}\\) d. \\(\\sqrt{36}\\) e. \\(\\sqrt{5}\\) f. \\(\\sqrt [ 3 ]{ 27 }\\) g. \u03c0=3.141592\u2026 h. Order the numbers in parts (a)\u2013(g) from least to greatest, and place them on a number line. Question 4. b. Which is greater, 4 or \\(\\sqrt{26}\\) ? c. Which is greater, \\(\\sqrt [ 3 ]{ 64 }\\) or \\(\\sqrt{16}\\) ? d. Which is greater, \\(\\sqrt [ 3 ]{ 125 }\\)5 or \\(\\sqrt{30}\\) ? e. Which is greater, -7 or –\\(\\sqrt{42}\\) ? f. Put the numbers 9, \\(\\sqrt{52}\\) , and \\(\\sqrt [ 3 ]{ 216 }\\) in order from least to greatest. Explain how you know which order to put them in. Question 5. b. Explain how you know where to place \\(\\sqrt{5}\\) on the number line. c. How could you improve the accuracy of your estimate? Question 6. b. x3<\/sup>=1000 c. 17 + x2 <\/sup>= 42 d. x3 <\/sup>+ 3x – 9 = x – 1 + 2x e. The cube shown has a volume of 216 cm3<\/sup>. ii. Solve the equation, and explain how you solved it. Engage NY Eureka Math 8th Grade Module 7 Mid Module Assessment Answer Key Eureka Math Grade 8 Module 7 Mid Module Assessment Task Answer Key Question 1. a. What is the decimal expansion of the number ? Is the number rational or irrational? Explain. Answer: = 5.000…. The number is a rational number. Rational numbers … Read more<\/a><\/p>\n","protected":false},"author":3,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"spay_email":""},"categories":[6],"tags":[],"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/posts\/84937"}],"collection":[{"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/comments?post=84937"}],"version-history":[{"count":1,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/posts\/84937\/revisions"}],"predecessor-version":[{"id":91465,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/posts\/84937\/revisions\/91465"}],"wp:attachment":[{"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/media?parent=84937"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/categories?post=84937"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/tags?post=84937"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\na. Write \\(0 . \\overline{345}\\) as a fraction.
\nAnswer:
\nLet x be \\(0 . \\overline{345}\\)
\n103<\/sup>x = 103<\/sup>(\\(0 . \\overline{345}\\))
\n1000 x = 1000(\\(0 . \\overline{345}\\))
\n1000x = \\(345 . \\overline{345}\\)
\n1000x = 345 + x
\n1000x – x = 345 + x – x
\n999x = 345
\nx = \\(\\frac{345}{999}\\) = \\(\\frac{115}{333}\\)<\/p>\n
\nAnswer:
\nLet x be \\(2.8 \\overline{40}\\)
\n10x = \\(2.8 \\overline{40}\\)
\n10x = 28 + \\(0. \\overline{40}\\)
\n10x = \\(\\frac{(28)(99)+40}{99}\\)
\nx = \\(\\frac{2812}{99}\\)(\\(\\frac{1}{10}\\))
\nx = \\(\\frac{2812}{990}\\) = \\(\\frac{1406}{495}\\)<\/p>\n
\n102<\/sup>y = 102<\/sup> (\\(0. \\overline{40}\\))
\n100y = \\(40. \\overline{40}\\)
\n100y = 40 + y
\n100y – y = 40 + y – y
\n99y = 40
\ny = \\(\\frac{40}{99}\\)
\n\\(2.8 \\overline{40}\\) = \\(\\frac{2812}{990}\\) = \\(\\frac{1406}{495}\\)<\/p>\n
\nAnswer:
\nNo, I do not agree with brandon. The decimal 0.66 is equal to the fraction \\(\\frac{66}{100}\\) = \\(\\frac{33}{50}\\), not \\(\\frac{2}{3}\\). Also, the number \\(\\frac{2}{3}\\) is equal to the infinit decimal \\(0 . \\overline{6}\\). The number 0.66 is a finite decimal. Therefore, 0.66 and \\(\\frac{2}{3}\\) are not equivlent.<\/p>\n
\nAnswer:
\nThe number \\(\\sqrt{33}\\) is between 5 and 6 because
\n52<\/sup> < (\\(\\sqrt{33}\\))2<\/sup> < 62<\/sup><\/p>\n
\nAnswer:
\n(5.25)2<\/sup> = 27.5625
\nSince \\(\\sqrt{x}\\) is the square root of x, then x2<\/sup> will give me the integer that belongs in the square root.
\n(5.25)2<\/sup> = 27.5625, which is closest to the integer 28.<\/p>\n
\nIdentify each of the following numbers as rational or irrational. If the number is irrational, explain how you know.
\na. \\(\\sqrt{29}\\)
\nAnswer:
\nIrrational because 29 is not a perfect square and \\(\\sqrt{29}\\) has an infinite decimal expansion that does not repeat.<\/p>\n
\nAnswer:
\nRational<\/p>\n
\nAnswer:
\nRational<\/p>\n
\nAnswer:
\nRational<\/p>\n
\nAnswer:
\nIrrational because 5 is not a perfect square and \\(\\sqrt{5}\\) has an infinite decimal expansion that does not repeat.<\/p>\n
\nAnswer:
\nRational<\/p>\n
\nAnswer:
\nIrrational because pi has a decimal expansion that does not repeat.<\/p>\n
\nAnswer:
\n\\(\\sqrt{29}\\) : 52<\/sup> < (\\(\\sqrt{29}\\))2<\/sup> < 62<\/sup>, 5.32<\/sup> < (\\(\\sqrt{29}\\))2<\/sup> < 5.42<\/sup>, 5.382<\/sup> < (\\(\\sqrt{29}\\))2<\/sup> < 5.392<\/sup>
\n![\"Engage](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/04\/Engage-NY-Math-8th-Grade-Module-7-Mid-Module-Assessment-Answer-Key-4.png\")
<\/p>\n
\nCircle the greater number in each of the pairs (a)\u2013(e) below.
\na. Which is greater, 8 or \\(\\sqrt{60}\\) ?
\nAnswer:
\n8<\/p>\n
\nAnswer:
\n\\(\\sqrt{26}\\)<\/p>\n
\nAnswer:
\nThe numbers are equal: \\(\\sqrt [ 3 ]{ 64 }\\) = 4, \\(\\sqrt{16}\\) = 4<\/p>\n
\nAnswer:
\n\\(\\sqrt{30}\\)<\/p>\n
\nAnswer:
\n–\\(\\sqrt{42}\\)<\/p>\n
\nAnswer:
\n\\(\\sqrt{52}\\) is between 7 and 8
\n\\(\\sqrt [ 3 ]{ 216 }\\) = 6
\nIn order from least to greatest:
\n\\(\\sqrt [ 3 ]{ 216 }\\), \\(\\sqrt{52}\\), 9<\/p>\n
\n![\"Engage](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/04\/Engage-NY-Math-8th-Grade-Module-7-Mid-Module-Assessment-Answer-Key-1.png\")
\nAnswer:
\n![\"Engage](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/04\/Engage-NY-Math-8th-Grade-Module-7-Mid-Module-Assessment-Answer-Key-5.png\")
\na. Between which two labeled points on the number line would \\(\\sqrt{5}\\) be located?
\nAnswer:
\nThe number \\(\\sqrt{5}\\) is between 2.2 and 2.3.<\/p>\n
\nAnswer:
\nI knew that \\(\\sqrt{5}\\) was between 2 and 3 but closer to 2. So next, I checked intervals of tenths beginning with 2.0 to 2.1. The interval that \\(\\sqrt{5}\\) fit between was 2.2 and 2.3 because 2.22<\/sup> < (\\(\\sqrt{5}\\))2<\/sup> < 2.32<\/sup>, 4.84 < 5 < 5.29.<\/p>\n
\nAnswer:
\nTo improve the estimate, I would have to continue with the method of rational approximation to determine which interval of hundreths \\(\\sqrt{5}\\) fits between. Once I knew the interval of hundreths, I would check the interval of thousandths, and so on.
\n\\(\\sqrt{5}\\) : 22<\/sup> < (\\(\\sqrt{5}\\))2<\/sup> < 32<\/sup>, 2.22<\/sup> < (\\(\\sqrt{5}\\))2<\/sup> < 2.32<\/sup> 4 < 5 < 9
\n4.84 < 5 < 5.29<\/p>\n
\nDetermine the positive solution for each of the following equations.
\na. 121 = x2<\/sup>
\nAnswer:
\n\\(\\sqrt{121}\\) = \\(\\sqrt{x^{2}}\\)
\n11 = x
\n112<\/sup> = 121
\n121 = 121<\/p>\n
\nAnswer:
\n\\(\\sqrt[3]{x^{3}}\\) = \\(\\sqrt [ 3 ]{ 1000 }\\)
\nx = 10
\n103<\/sup> = 1000
\n1000 = 1000<\/p>\n
\nAnswer:
\n17 – 17 + x2<\/sup> = 42 – 17
\nx2<\/sup> = 25
\n\\(\\sqrt{x^{2}}\\) = \\(\\sqrt{25}\\)
\nx = 5
\n17 + 52<\/sup> = 42
\n17 + 25 = 42
\n42 = 42<\/p>\n
\nAnswer:
\nx3<\/sup> + 3x – 3x – 9 = x – 1 + 2x – 3x
\nx3<\/sup> – 9 = -1
\nx3<\/sup> – 9 + 9 = -1 + 9
\nx3<\/sup> = 8
\n\\(\\sqrt[3]{x^{3}}\\) = \\(\\sqrt [ 3 ]{ 8 }\\)
\nx = 2
\n23<\/sup> + (3)(2) – 9 = 2 – 1 + (2)(2)
\n8 + 6 – 9 = 2 – 1 + 4
\n14 – 9 = 1 + 4
\n5 = 5<\/p>\n
\ni. Write an equation that could be used to determine the length, l, of one side.
\n![\"Engage](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/04\/Engage-NY-Math-8th-Grade-Module-7-Mid-Module-Assessment-Answer-Key-2.png\")
\nAnswer:
\nV = l3<\/sup>
\n216 = l3<\/sup><\/p>\n
\nAnswer:
\n216 = l3<\/sup>
\n\\(\\sqrt [ 3 ]{ 216 }\\) = \\(\\sqrt[3]{l^{3}}\\)
\n6 = l
\nThe length of one side is 6 cm.
\nTo solve the equation, I had to take the cube root of both sides of the equation. The cube root of l3<\/sup>, \\(\\sqrt[3]{l^{3}}\\), is l. The cube root of 216, \\(\\sqrt [ 3 ]{ 216 }\\), is 6 because 63<\/sup> = 216.
\nTherefore, the length of one side of the cube is 6 cm.<\/p>\n","protected":false},"excerpt":{"rendered":"