Students can use the **Spectrum Math Grade 8 Answer Key** **Chapter 2 Lesson 2.7 Comparing and Ordering Irrational Numbers** as a quick guide to resolve any of their doubts.

## Spectrum Math Grade 8 Chapter 2 Lesson 2.7 Comparing and Ordering Irrational Numbers Answers Key

Rational and irrational numbers can be compared by approximating their value and placing them along a number line.

Place these numbers on a number line: \(\sqrt{5}\), 2.5, \(\sqrt{3}\)

**Put the values below in order from least to greatest along a number line.**

Question 1.

π^{2}, 10, \(\sqrt{75}\)

Answer:

Rational and irrational numbers can be compared by approximating their value and placing them along a number line.

Question 2.

\(\sqrt{7}\), \(\frac{\sqrt{7}}{2}\), 2

Answer:

Rational and irrational numbers can be compared by approximating their value and placing them along a number line.

Question 3.

\(\sqrt{10}\), 3.5, 2^{2}

Answer:

Rational and irrational numbers can be compared by approximating their value and placing them along a number line.

Question 4.

4, \(\sqrt{15}\), 5.2

Answer:

Rational and irrational numbers can be compared by approximating their value and placing them along a number line.

Question 5.

\(\frac{1}{3}\), \(\sqrt{1}\), 0.45

Answer:

Rational and irrational numbers can be compared by approximating their value and placing them along a number line.

Question 6.

\(\sqrt{72}\), 9, 8^{2}

Answer:

Rational and irrational numbers can be compared by approximating their value and placing them along a number line.

Expressions and equations containing irrational numbers can be approximated by testing values.

Compare using <, >, or =.

\(\sqrt{3}\) + 5 ____ 3 + \(\sqrt{5}\)

\(\sqrt{3}\) is between 1 and 2, so use 1.5.

\(\sqrt{5}\) is between 2 and 3, so use 2.5.

(1.5) + 5 ____ 3 + (2.5)

6.5 > 5.5

**Approximate the value of each expression and then compare using <, >, or =.**

Question 1.

a. \(\sqrt{10}\) + 2 _____ 10 + \(\sqrt{2}\)

Approximation: ____________

Answer: \(\sqrt{10}\) + 2 < 10 + \(\sqrt{2}\)

\(\sqrt{10}\) + 2 _____ 10 + \(\sqrt{2}\)

\(\sqrt{10}\) is between 3 and 4, so use 3.5.

\(\sqrt{2}\) is between 1 and 2, so use 1.5.

3.5 + 2 _________ 10 + 1.5

5.5 < 11.5

b. 4 + \(\sqrt{2}\) _____ \(\sqrt{4}\) + 2

Approximation: ____________

Answer: 4 + \(\sqrt{2}\) > \(\sqrt{4}\) + 2

4 + \(\sqrt{2}\) _____ \(\sqrt{4}\) + 2

\(\sqrt{2}\) is between 1 and 2, so use 1.5.

\(\sqrt{4}\) is 2.

4 + 1.5 _____2 + 2

5.5 > 4

Question 2.

a. 12 + \(\sqrt{6}\) _____ \(\sqrt{12}\) + 6

Approximation: ____________

Answer: 12 + \(\sqrt{6}\) < \(\sqrt{12}\) + 6

12 + \(\sqrt{6}\) _____ \(\sqrt{12}\) + 6

\(\sqrt{6}\) is between 2 and 3, so use 2.5.

\(\sqrt{12}\) is between 3 and 4, so use 3.5.

12 +2.5 _________ 3.5+6

14.5 < 9.5

b. \(\sqrt[3]{8}\) + 6 _____ 8 + \(\sqrt[3]{6}\)

Approximation: ____________

Answer: \(\sqrt[3]{8}\) + 6 < 8 + \(\sqrt[3]{6}\)

\(\sqrt[3]{8}\) + 6 _____ 8 + \(\sqrt[3]{6}\)

\(\sqrt[3]{8}\) is 2.

\(\sqrt[3]{6}\) is 1 and 2, so use 1.5.

2 + 6 ______ 8+1.5

8 < 9.5

Question 3.

a. 15 + \(\sqrt{12}\) _____ \(\sqrt{15}\) + 12

Approximation: ____________

Answer: 15 + \(\sqrt{12}\) > \(\sqrt{15}\) + 12

15 + \(\sqrt{12}\) _____ \(\sqrt{15}\) + 12

\(\sqrt{12}\) is between 3 and 4, so use 3.5.

\(\sqrt{15}\) is between 3 and 4, so use 3.5.

15 + 3.5 ________ 3.5+12

18.5 > 15.5

b. \(\sqrt{7}\) + 3 _____ 7 + \(\sqrt{3}\)

Approximation: ____________

Answer: \(\sqrt{7}\) + 3 < 7 + \(\sqrt{3}\)

\(\sqrt{7}\) + 3 _____ 7 + \(\sqrt{3}\)

\(\sqrt{7}\) is between 2 and 3, so use 2.5.

\(\sqrt{3}\) is between 1 and 2, so use 1.5.

2.5 + 3 ________ 7+1.5

5.5 < 8.5

Question 4.

a. 4 + \(\sqrt[3]{7}\) _____ \(\sqrt[3]{4}\) + 7

Approximation: ____________

Answer: 4 + \(\sqrt[3]{7}\) < \(\sqrt[3]{4}\) + 7

4 + \(\sqrt[3]{7}\) _____ \(\sqrt[3]{4}\) + 7

\(\sqrt[3]{7}\) is between 1 and 2, so use 1.5.

\(\sqrt[3]{4}\) is between 1 and 2, so use 1.5.

4 + 1.5 ________ 1.5 + 7

5.5 < 8.5

b. \(\sqrt{3}\) + 5 _____ 3 + \(\sqrt{5}\)

Approximation: ____________

Answer: \(\sqrt{3}\) + 5 > 3 + \(\sqrt{5}\)

\(\sqrt{3}\) + 5 _____ 3 + \(\sqrt{5}\)

\(\sqrt{3}\) is between 1 and 2, so use 1.5.

\(\sqrt{5}\) is between 2 and 3, so use 2.5.

1.5 + 5 ________ 3 + 2.5

6.5 > 5.5