Exploratory Challenge\/Exercises 1\u201311<\/p>\n
Exercise 1.
\nRodney thinks that \\(\\sqrt [ 3 ]{ 64 }\\) is greater than \\(\\frac{17}{4}\\). Sam thinks that \\(\\frac{17}{4}\\) is greater. Who is right and why?
\n\\(\\sqrt [ 3 ]{ 64 }\\) = \\(\\sqrt[3]{4^{3}}\\) = 4
\nand
\n\\(\\frac{17}{4}\\) = \\(\\frac{16}{4}\\) + \\(\\frac{1}{4}\\)
\n= 4 + \\(\\frac{1}{4}\\)
\n= 4 \\(\\frac{1}{4}\\)
\nWe see that \\(\\sqrt [ 3 ]{ 64 }\\) < \\(\\frac{17}{4}\\). Sam is correct.
\n![\"Engage](\"https:\/\/ccssanswers.com\/wp-content\/uploads\/2021\/04\/Engage-NY-Math-Grade-8-Module-7-Lesson-13-Exercise-Answer-Key-1.png\")
<\/p>\n
Exercise 2.
\nWhich number is smaller, \\(\\sqrt [ 3 ]{ 27 }\\) or 2.89? Explain.
\nAnswer:
\n\\(\\sqrt [ 3 ]{ 27 }\\) = \\(\\sqrt[3]{3^{3}}\\) = 3
\nWe see that 2.89 is smaller than \\(\\sqrt [ 3 ]{ 27 }\\).<\/p>\n
Exercise 3.
\nWhich number is smaller, \\(\\sqrt{121}\\) or \\(\\sqrt [ 3 ]{ 125 }\\)? Explain.
\nAnswer:
\n\\(\\sqrt{121}\\) = \\(\\sqrt{11^{2}}\\) = 11
\n\\(\\sqrt [ 3 ]{ 125 }\\) = \\(\\sqrt[3]{5^{3}}\\) = 5
\nWe see that \\(\\sqrt [ 3 ]{ 125 }\\) is smaller than \\(\\sqrt{121}\\).<\/p>\n
Exercise 4.
\nWhich number is smaller, \\(\\sqrt{49}\\) or \\(\\sqrt [ 3 ]{ 216 }\\)? Explain.
\nAnswer:
\n\\(\\sqrt{49}\\) = \\(\\sqrt{7^{2}}\\) = 7
\n\\(\\sqrt [ 3 ]{ 216 }\\) = \\(\\sqrt[3]{6^{3}}\\) = 6
\nWe see that \\(\\sqrt [ 3 ]{ 216 }\\) is smaller than \\(\\sqrt{49}\\).<\/p>\n
Exercise 5. Exercise 6. Exercise 7. Exercise 8. The number \\(\\sqrt{2}\\) is between 1 and 2 because 12<\/sup> < 2 < 22<\/sup>. The number \\(\\sqrt{2}\\) is between 1.4 and 1.5 because Exercise 9. Exercise 10. Alternately: Exercise 11. Question 1. Question 2. Question 3. The number \\(\\sqrt{87}\\) is between 9 and 10 because 92<\/sup> < 87 < 102<\/sup>. The number \\(\\sqrt{87}\\) is between 9.3 and 9.4 because 9.32<\/sup> < 87 < 9.42<\/sup>. The number \\(\\sqrt{87}\\) is between 9.32 and 9.33 because 9.322<\/sup> < 87 < 9.332<\/sup>. Since \\(\\sqrt{87}\\) < 9.3, then \\(\\sqrt{87}\\) < \\(\\frac{929}{99}\\). The fraction \\(\\frac{929}{99}\\) is larger.<\/p>\n Question 4. Question 5. Question 6. The numbers \\(\\sqrt{110}\\), \\(\\sqrt{115}\\), and \\(\\sqrt{120}\\) are all between 10 and 11 because when squared, their value falls between 102<\/sup> and 112<\/sup>. The number \\(\\sqrt{110}\\) is between 10.4 and 10.5 because 10.42<\/sup> < 110 < 10.52<\/sup>. The number \\(\\sqrt{115}\\) is between 10.7 and 10.8 because 10.72<\/sup> < 115 < 10.82<\/sup>. The number \\(\\sqrt{120}\\) is between 10.9 and 11 because 10.92<\/sup> < 120 < 112<\/sup>. Question 7. The triangle on the left has the longer hypotenuse. It is approximately 0.21 units longer than the hypotenuse of the triangle on the right. Question 1. Engage NY Eureka Math 8th Grade Module 7 Lesson 13 Answer Key Eureka Math Grade 8 Module 7 Lesson 13 Exploratory Challenge\/Exercise Answer Key Exploratory Challenge\/Exercises 1\u201311 Exercise 1. Rodney thinks that is greater than . Sam thinks that is greater. Who is right and why? = = 4 and = + = 4 + … Read more<\/a><\/p>\n","protected":false},"author":3,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[6],"tags":[],"jetpack_sharing_enabled":true,"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/posts\/83265"}],"collection":[{"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/comments?post=83265"}],"version-history":[{"count":1,"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/posts\/83265\/revisions"}],"predecessor-version":[{"id":137282,"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/posts\/83265\/revisions\/137282"}],"wp:attachment":[{"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/media?parent=83265"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/categories?post=83265"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/tags?post=83265"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\nWhich number is greater, \\(\\sqrt{50}\\) or \\(\\frac{319}{45}\\)? Explain.
\nAnswer:
\nStudents may use any method to determine the decimal expansion of the fraction.
\nThe number \\(\\frac{319}{45}\\) is equal to\\(7.0 \\overline{8}\\).
\nThe number \\(\\sqrt{50}\\) is between 7 and 8 because 72<\/sup> < 50 < 82<\/sup>. The number \\(\\sqrt{50}\\) is between 7.0 and 7.1 because 72<\/sup> < 50 < 7.12<\/sup>. The number \\(\\sqrt{50}\\) is between 7.07 and 7.08 because 7.072<\/sup> < 50 < 7.082<\/sup>. The approximate decimal value of \\(\\sqrt{50}\\) is 7.07. Since 7.07 < \\(7.0 \\overline{8}\\), then \\(\\sqrt{50}\\) < \\(\\frac{319}{45}\\); therefore, the fraction \\(\\frac{319}{45}\\) is greater than \\(\\sqrt{50}\\) .
\nAlternately: (\\(\\sqrt{50}\\))2<\/sup> = 50 and (\\(\\frac{319}{45}\\))2<\/sup> = 101761\/2025>101250\/2025 = 50. So, 319\/45 must be larger.<\/p>\n
\nWhich number is greater, \\(\\frac{5}{11}\\) or \\(0. \\overline{4}\\)? Explain.
\nAnswer:
\nStudents may use any method to determine the decimal expansion of the fraction.
\nThe number \\(\\frac{5}{11}\\) is equal to \\(0. \\overline{45}\\). Since 0.44444\u2026 < 0.454545\u2026, then \\(0. \\overline{4}\\) < \\(\\frac{5}{11}\\); therefore, the fraction \\(\\frac{5}{11}\\) is greater than 0.4 \u0305.
\nAlternately: 0.444\u2026 = \\(\\frac{4}{9}\\), and we can compare the fractions \\(\\frac{4}{9}\\) and \\(\\frac{5}{11}\\) using their equivalents, \\(\\frac{44}{99}\\) and \\(\\frac{45}{99}\\) to see that \\(\\frac{5}{11}\\) is larger.<\/p>\n
\nWhich number is greater, \\(\\sqrt{38}\\) or \\(\\frac{154}{25}\\)? Explain.
\nAnswer:
\nStudents may use any method to determine the decimal expansion of the fraction.
\n\\(\\frac{154}{25}\\) = \\(\\frac{154 \\times 4}{25 \\times 4} = \\frac{616}{10^{2}}\\) = 6.16
\nThe number \\(\\sqrt{38}\\) is between 6 and 7 because 62<\/sup> < 38 < 72<\/sup>. The number \\(\\sqrt{38}\\) is between 6.1 and 6.2 because
\n6.12<\/sup> < 38 < 6.22<\/sup>. The number \\(\\sqrt{38}\\) is between 6.16 and 6.17 because 6.162<\/sup> < 38 < 6.172<\/sup>. Since \\(\\sqrt{38}\\) is greater than 6.16, then \\(\\sqrt{38}\\) is greater than 154\/25.
\nAlternately: (\\(\\sqrt{38}\\))2<\/sup> = 38 and (\\(\\frac{154}{25}\\))2<\/sup> = \\(\\frac{23716}{625}\\) < \\(\\frac{23750}{625}\\) = 38. So, \\(\\sqrt{38}\\) must be larger.<\/p>\n
\nWhich number is greater, \\(\\sqrt{2}\\) or \\(\\frac{15}{9}\\)? Explain.
\nAnswer:
\nStudents may use any method to determine the decimal expansion of the fraction.
\nThe number \\(\\frac{15}{9}\\) is equal to\\(1 . \\overline{6}\\).<\/p>\n
\n1.42<\/sup> < 2 < 1.52<\/sup>. Therefore, \\(\\sqrt{2}\\) < \\(\\frac{15}{9}\\); the fraction \\(\\frac{15}{9}\\) is greater.
\nAlternately: (\\(\\sqrt{2}\\))2<\/sup> = 2 and (\\(\\frac{15}{9}\\))2<\/sup> = (\\(\\frac{5}{3}\\))2<\/sup> = \\(\\frac{28}{9}\\)>2. So, \\(\\frac{15}{9}\\) must be larger.<\/p>\n
\nPlace each of the following numbers at its approximate location on the number line: \\(\\sqrt{25}\\), \\(\\sqrt{28}\\), \\(\\sqrt{30}\\), \\(\\sqrt{32}\\), \\(\\sqrt{35}\\), and \\(\\sqrt{36}\\).
\nAnswer:
\nThe solutions are shown in red:
\n![\"Engage](\"https:\/\/ccssanswers.com\/wp-content\/uploads\/2021\/04\/Engage-NY-Math-Grade-8-Module-7-Lesson-13-Exercise-Answer-Key-2.png\")
\nThe number \\(\\sqrt{25}\\) = \\(\\sqrt{5^{2}}\\) = 5.
\nThe numbers \\(\\sqrt{28}\\),\\(\\sqrt{30}\\),\\(\\sqrt{32}\\), and \\(\\sqrt{35}\\) are between 5 and 6. The number \\(\\sqrt{28}\\) is between 5.2 and 5.3 because 5.22<\/sup> < 28 < 5.32<\/sup>. The number \\(\\sqrt{30}\\) is between 5.4 and 5.5 because 5.42<\/sup> < 30 < 5.52<\/sup>. The number \\(\\sqrt{32}\\) is between 5.6 and 5.7 because 5.62<\/sup> < 32 < 5.72<\/sup>. The number \\(\\sqrt{35}\\) is between 5.9 and 6.0 because
\n5.92<\/sup> < 35 < 62<\/sup>.
\nThe number\\(\\sqrt{36}\\) = \\(\\sqrt{6^{2}}\\) = 6.<\/p>\n
\nChallenge: Which number is larger, \\(\\sqrt{5}\\) or \\(\\sqrt [ 3 ]{ 11 }\\)?
\nAnswer:
\nThe number \\(\\sqrt{5}\\) is between 2 and 3 because 22<\/sup> < 5 < 32<\/sup>. The number \\(\\sqrt{5}\\) is between 2.2 and 2.3 because
\n2.22<\/sup> < 5 < 2.32<\/sup>. The number \\(\\sqrt{5}\\) is between 2.23 and 2.24 because 2.232<\/sup> < 5 < 2.242<\/sup>. The number \\(\\sqrt{5}\\) is between 2.236 and 2.237 because 2.2362<\/sup> < 5 < 2.2372<\/sup>. The decimal expansion of \\(\\sqrt{5}\\) is approximately 2.236\u2026.
\nThe number \\(\\sqrt [ 3 ]{ 11 }\\) is between 2 and 3 because 2^3 < 11 < 3^3. The number \\(\\sqrt [ 3 ]{ 11 }\\) is between 2.2 and 2.3 because 2.2^3 < 11 < 2.3^3. The number \\(\\sqrt [ 3 ]{ 11 }\\) is between 2.22 and 2.23 because 2.22^3 < 11 < 2.23^3. The decimal expansion of \\(\\sqrt [ 3 ]{ 11 }\\) is approximately 2.22\u2026. Since 2.22″\u2026” < 2.236″\u2026” , then \\(\\sqrt [ 3 ]{ 11 }\\) < \\(\\sqrt{5}\\); therefore, \\(\\sqrt{5}\\) is larger.<\/p>\n
\n(\\(\\sqrt [ 3 ]{ 11 }\\))^6 = 112<\/sup> = 121
\n(\\(\\sqrt{5}\\))^6 = 5^3 = 125
\nWe see that \\(\\sqrt{5}\\) must be larger.<\/p>\n
\nA certain chessboard is being designed so that each square has an area of 3 in2<\/sup>. What is the length of one edge of the board rounded to the tenths place? (A chessboard is composed of 64 squares as shown.)
\n![\"Engage](\"https:\/\/ccssanswers.com\/wp-content\/uploads\/2021\/04\/Engage-NY-Math-Grade-8-Module-7-Lesson-13-Exercise-Answer-Key-3.png\")
\nAnswer:
\nThe area of one square is 3 in2<\/sup>. So, if x is the length of one side of one square,
\nx2<\/sup> = 3
\n\\(\\sqrt{x^{2}}\\) = \\(\\sqrt{3}\\)
\nx = \\(\\sqrt{3}\\).
\nThere are 8 squares along one edge of the board, so the length of one edge is 8\u00d7\\(\\sqrt{3}\\). The number \\(\\sqrt{3}\\) is between 1 and 2 because
\n12<\/sup> < 3 < 22<\/sup>. The number \\(\\sqrt{3}\\) is between 1.7 and 1.8 because 1.72<\/sup> < 3 < 1.82<\/sup>. The number \\(\\sqrt{3}\\) is between 1.73 and 1.74 because 1.732<\/sup> < 3 < 1.742<\/sup>. The number \\(\\sqrt{3}\\) is approximately 1.73. So, the length of one edge of the chessboard is about
\n8\u00d71.73 inches, which is approximately 13.8 in.
\nNote: Some students may determine the total area of the board, 64\u00d73 = 192, and then determine the approximate value of \\(\\sqrt{192}\\) as 13.8 to answer the question.<\/p>\nEureka Math Grade 8 Module 7 Lesson 13 Problem Set Answer Key<\/h3>\n
\nWhich number is smaller, \\(\\sqrt [ 3 ]{ 343 }\\) or \\(\\sqrt{48}\\) ? Explain.
\nAnswer:
\n\u221a(3&343) = \\(\\sqrt[3]{7^{3}}\\) = 7
\nThe number \\(\\sqrt{48}\\) is between 6 and 7 but definitely less than 7. Therefore, \\(\\sqrt{48}\\) < \\(\\sqrt [ 3 ]{ 343 }\\), and \u221a\\(\\sqrt{48}\\) is smaller.<\/p>\n
\nWhich number is smaller, \\(\\sqrt{100}\\) or \\(\\sqrt [ 3 ]{ 1000 }\\)? Explain.
\nAnswer:
\n\\(\\sqrt{100}\\) = \\(\\sqrt{10^{2}}\\) = 10
\n\\(\\sqrt [ 3 ]{ 1000 }\\) = \\(\\sqrt[3]{10^{3}}\\) = 10
\nThe numbers \\(\\sqrt{100}\\) and \\(\\sqrt [ 3 ]{ 1000 }\\) are equal because both are equal to 10.<\/p>\n
\nWhich number is larger, \\(\\sqrt{87}\\) or \\(\\frac{929}{99}\\)? Explain.
\nAnswer:
\nStudents may use any method to compute the first few decimal places of a fraction.
\nThe number \\(\\frac{929}{99}\\) is equal to \\(9 . \\overline{38}\\).<\/p>\n
\nWhich number is larger, \\(\\frac{9}{13}\\) or \\(0 . \\overline{692}\\)? Explain.
\nAnswer:
\nStudents may use any method to compute the first few decimal places of a fraction.
\nThe number \\(\\frac{9}{13}\\) is equal to \\(0 . \\overline{692307}\\). Since 0.692307\u2026 < 0.692692\u2026, then we see that 9\/13 < \\(0 . \\overline{692}\\).
\nThe decimal \\(0 . \\overline{692}\\) is larger.<\/p>\n
\nWhich number is larger, 9.1 or \\(\\sqrt{82}\\)? Explain.
\nAnswer:
\nThe number \\(\\sqrt{82}\\) is between 9 and 10 because 92<\/sup> < 82 < 102<\/sup>. The number \\(\\sqrt{82}\\) is between 9.0 and 9.1 because 9.02<\/sup> < 82 < 9.12<\/sup>. Since \\(\\sqrt{82}\\) < 9.1, then the number 9.1 is larger than the number \\(\\sqrt{82}\\).<\/p>\n
\nPlace each of the following numbers at its approximate location on the number line: \\(\\sqrt{144}\\), \\(\\sqrt [ 3 ]{ 1000 }\\), \\(\\sqrt{130}\\), \\(\\sqrt{110}\\), \\(\\sqrt{120}\\), \\(\\sqrt{115}\\), and \\(\\sqrt{133}\\). Explain how you knew where to place the numbers.
\n![\"Eureka](\"https:\/\/ccssanswers.com\/wp-content\/uploads\/2021\/04\/Eureka-Math-8th-Grade-Module-7-Lesson-13-Problem-Set-Answer-Key-1.png\")
\nAnswer:
\nThe solutions are shown in red:
\n![\"Eureka](\"https:\/\/ccssanswers.com\/wp-content\/uploads\/2021\/04\/Eureka-Math-8th-Grade-Module-7-Lesson-13-Problem-Set-Answer-Key-2.png\")
\nThe number \\(\\sqrt{144}\\) = \\(\\sqrt{12^{2}}\\) = 12.
\nThe number \\(\\sqrt [ 3 ]{ 1000 }\\) = \\(\\sqrt[3]{10^{3}}\\) = 10.<\/p>\n
\nThe numbers \u221a130 and \\(\\sqrt{133}\\) are between 11 and 12 because when squared, their value falls between 112<\/sup> and 122<\/sup>. The number \u221a130 is between 11.4 and 11.5 because 11.42<\/sup> < 130 < 11.52<\/sup>. The number \\(\\sqrt{133}\\) is between 11.5 and 11.6 because 11.52<\/sup> < 133 < 11.62<\/sup>.<\/p>\n
\nWhich of the two right triangles shown below, measured in units, has the longer hypotenuse? Approximately how much longer is it?
\n![\"Eureka](\"https:\/\/ccssanswers.com\/wp-content\/uploads\/2021\/04\/Eureka-Math-8th-Grade-Module-7-Lesson-13-Problem-Set-Answer-Key-3.png\")
\nAnswer:
\nLet x represent the length of the hypotenuse of the triangle on the left.
\n72<\/sup> + 22<\/sup> = x2<\/sup>
\n49 + 4 = x2<\/sup>
\n53 = x2<\/sup>
\n\\(\\sqrt{53}\\) = x
\nThe number \\(\\sqrt{53}\\) is between 7 and 8 because 72<\/sup> < 53 < 82<\/sup>. The number \\(\\sqrt{53}\\) is between 7.2 and 7.3 because 7.22<\/sup> < 53 < 7.32<\/sup>. The number \\(\\sqrt{53}\\) is between 7.28 and 7.29 because 7.282<\/sup> < 53 < 7.292<\/sup>. The approximate decimal value of \\(\\sqrt{53}\\) is 7.28.
\nLet y represent the length of the hypotenuse of the triangle on the right.
\n52<\/sup> + 52<\/sup> = y2<\/sup>
\n25 + 25 = y2<\/sup>
\n50 = y2<\/sup>
\n\\(\\sqrt{50}\\) = y
\nThe number \\(\\sqrt{50}\\) is between 7 and 8 because 72<\/sup> < 50 < 82<\/sup>. The number \\(\\sqrt{50}\\) is between 7.0 and 7.1 because 7.02<\/sup> < 50 < 7.12<\/sup>. The number \\(\\sqrt{50}\\) is between 7.07 and 7.08 because 7.072<\/sup> < 50 < 7.082<\/sup>. The approximate decimal value of \\(\\sqrt{50}\\) is 7.07.<\/p>\n
\nNote: Based on their experience, some students may reason that \\(\\sqrt{50}\\) < \\(\\sqrt{53}\\). To answer completely, students must determine the decimal expansion to approximate how much longer one hypotenuse is than the other.<\/p>\nEureka Math Grade 8 Module 7 Lesson 13 Exit Ticket Answer Key<\/h3>\n
\nPlace each of the following numbers at its approximate location on the number line: \\(\\sqrt{12}\\), \\(\\sqrt{16}\\), \\(\\frac{20}{6}\\), \\(3 . \\overline{53}\\), and \\(\\sqrt [ 3 ]{ 27 }\\).
\n![\"Eureka](\"https:\/\/ccssanswers.com\/wp-content\/uploads\/2021\/04\/Eureka-Math-Grade-8-Module-7-Lesson-13-Exit-Ticket-Answer-Key-1.png\")
\nAnswer:
\nStudents may use any method to compute the first few decimal places of a fraction.
\nThe number \\(\\sqrt{12}\\) is between 3.4 and 3.5 since 3.42<\/sup> < 12 < 3.52<\/sup>.
\nThe number \\(\\sqrt{16}\\) = \\(\\sqrt{4^{2}}\\) = 4.
\nThe number \\(\\frac{20}{6}\\) is equal to \\(3 . \\overline{3}\\).
\nThe number \\(\\sqrt [ 3 ]{ 27 }\\) = \\(\\sqrt[3]{3^{3}}\\) = 3.
\nThe solutions are shown in red:
\n![\"Eureka](\"https:\/\/ccssanswers.com\/wp-content\/uploads\/2021\/04\/Eureka-Math-Grade-8-Module-7-Lesson-13-Exit-Ticket-Answer-Key-2.png\")
<\/p>\n","protected":false},"excerpt":{"rendered":"