Students can use the Spectrum Math Grade 8 Answer Key Chapter 2 Lesson 2.5 Comparing Rational and Irrational Numbers as a quick guide to resolve any of their doubts.
Spectrum Math Grade 8 Chapter 2 Lesson 2.5 Comparing Rational and Irrational Numbers Answers Key
Compare rational and irrational numbers by using a best guess for irrational numbers. \(\sqrt{3}\) < 2 This statement is true because \(\sqrt{3}\) is between 1 and 2. 5 > \(\sqrt{20}\) This statement is true because \(\sqrt{20}\) is between 4 and 5.
Compare using <, >, or =.
Question 1.
a. \(\sqrt{9}\) ______ π
Answer: \(\sqrt{9}\)< π
This statement is true because \(\sqrt{9}\) is 3 and value of π is 3.14.
b. 4.5 _____ \(\sqrt{25}\)
Answer: 4.5 < \(\sqrt{25}\)
This statement is true because \(\sqrt{25}\) is 5. As 4.5 is less than 5. Therefore 4.5 is less than \(\sqrt{25}\) .
c. 3.9 _____ \(\sqrt{10}\)
Answer: 3.9 > \(\sqrt{10}\)
This statement is true because \(\sqrt{10}\) is 3.16. As 3.16 is less than 3.9. Therefore, 3.9 is greater than
\(\sqrt{10}\).
Question 2.
a.
\(\sqrt{2}\) _____ 1
Answer: \(\sqrt{2}\) > 1
This statement is true because \(\sqrt{2}\) is 1.414. As 1.414 is greater than 1. Therefore, \(\sqrt{2}\) > 1
b.
\(3 \sqrt{\frac{8}{27}}\) ____ \(\frac{2}{3}\)
Answer: \(3 \sqrt{\frac{8}{27}}\) = \(\frac{2}{3}\)
This statement is true because \(3 \sqrt{\frac{8}{27}}\) is \(\frac{2}{3}\). Thereore, \(3 \sqrt{\frac{8}{27}}\) is equal to \(\frac{2}{3}\)
c. 1.1 ____ \(\sqrt{2}\)
Answer: 1.1 < \(\sqrt{2}\)
This statement is true because \(\sqrt{2}\) is 1.414. As 1.414 is greater than 1. Therefore, 1.1 is less than \(\sqrt{2}\).
Question 3.
a.
\(0 . \overline{66}\) _____ \(\frac{2}{3}\)
Answer: \(0 . \overline{66}\) = \(\frac{2}{3}\)
This statement is true because \(\frac{2}{3}\) is \(0 . \overline{66}\). Therefore, \(0 . \overline{66}\) = \(\frac{2}{3}\)
b.
\(\sqrt{8}\) _____ 3
Answer: \(\sqrt{8}\) < 3
This statement is true because \(\sqrt{8}\) is 2.82. As 2.82 is less than 3. Therefore, \(\sqrt{8}\) is less than 3.
c.
1 ______ \(\sqrt{\frac{16}{25}}\)
Answer: 1 > \(\sqrt{\frac{16}{25}}\)
This statement is true because \(\sqrt{\frac{16}{25}}\) is 0.8. As 1 is greater than 0.8. Therefore, 1 is greater than \(\sqrt{\frac{16}{25}}\).
Question 4.
a. \(\sqrt{36}\) ____ 6.5
Answer: \(\sqrt{36}\) < 6.5
This statement is true because \(\sqrt{36}\) is 6. As 6 is less than 6.5. Therefore, \(\sqrt{36}\)is less than 6.5.
b. 1 ____ \(0 . \overline{45}\)
Answer: 1 > \(0 . \overline{45}\)
This statement is true because 1 is always greater than \(0 . \overline{45}\)
c. \(\frac{3}{5}\) _____ \(\sqrt{\frac{9}{5}}\)
Answer: \(\frac{3}{5}\) < \(\sqrt{\frac{9}{5}}\)
This statement is true because \(\frac{3}{5}\) is 0.6 and \(\sqrt{\frac{9}{5}}\) is 1.34. As 1.34 is greater than 0.6. Therefore, \(\frac{3}{5}\) < \(\sqrt{\frac{9}{5}}\)
Question 5.
a.
\(\sqrt[3]{343}\) ____ 7.2
Answer: \(\sqrt[3]{343}\) < 7.2
this statement is true because \(\sqrt[3]{343}\) is 7. As 7 is less than 7.2. Therefore, \(\sqrt[3]{343}\) is less than 7.2.
b. \(0 . \overline{77}\) ____ \(\frac{7}{9}\)
Answer: \(0 . \overline{77}\) = \(\frac{7}{9}\)
This statement is true because \(\frac{7}{9}\) is \(0 . \overline{77}\). Therefore, \(0 . \overline{77}\) = \(\frac{7}{9}\)
c. 7 _____ \(\sqrt{52}\)
Answer: 7 < \(\sqrt{52}\)
This statement is true because \(\sqrt{52}\) is 7.2. As 7 is less than 7.2. Therefore, 7 is less than \(\sqrt{52}\).
Question 6.
a. \(\sqrt{5}\) ____ 4
Answer: \(\sqrt{5}\) < 4
This statement is true because \(\sqrt{5}\) is 2.23. As 2.23 is less than 4. Therefore, \(\sqrt{5}\) is less than 4.
b. \(\frac{3}{4}\) _____ \(0 . \overline{75}\)
Answer: \(\frac{3}{4}\) < \(0 . \overline{75}\)
This statement is true because \(\frac{3}{4}\) is 0.75. As 0.75 is less than \(0 . \overline{75}\). Therefore, \(\frac{3}{4}\) is less than \(0 . \overline{75}\)
c. 
____ 3.5
Answer: < 3.5
This statement is true because is 3.17. As 3.17 is less than 3.5. Therefore, is less than 3.5.
Question 7.
a. \(\frac{5}{10}\) _____ \(\sqrt{1}\)
Answer: \(\frac{5}{10}\) < \(\sqrt{1}\)
This statement is true because \(\frac{5}{10}\) is 0.5 and \(\sqrt{1}\) is 1. As 0.5 is less than 1. Therefore, \(\frac{5}{10}\) is less than \(\sqrt{1}\).
b. \(\sqrt[3]{6}\) ____ 2
Answer: \(\sqrt[3]{6}\) < 2
This statement is true because \(\sqrt[3]{6}\) is 0.5. As 0.5 is less than 2. Therefore, \(\sqrt[3]{6}\) is less than 2.
c. 1.4 ____ \(\sqrt{2}\)
Answer: 1.4 = \(\sqrt{2}\)
This statement is true because \(\sqrt{2}\) is 1.4. Therefore, 1.4 is equal to \(\sqrt{2}\).
Question 8.
a.
\(\sqrt[3]{\frac{27}{125}}\) ____ 0.6
Answer: \(\sqrt[3]{\frac{27}{125}}\) = 0.6
This statement is true because \(\sqrt[3]{\frac{27}{125}}\) is 0.6. Therefore \(\sqrt[3]{\frac{27}{125}}\) is equal to 0.6.
b.
\(\frac{1}{2}\) ____ 0.55
Answer: \(\frac{1}{2}\) < 0.55
This statement is true because \(\frac{1}{2}\) is 0.5. As 0.5 is less than 0.55. Therefore, \(\frac{1}{2}\) is less than 0.55.
c.
\(\sqrt[3]{18}\) ____ 2.5
Answer: \(\sqrt[3]{18}\) < 2.5
This statement is true because \(\sqrt[3]{18}\) is 0.408. As 0.408 is less than 2.5. Therefore, \(\sqrt[3]{18}\) is less than 2.5