Students can use the **Spectrum Math Grade 8 Answer Key** **Chapter 2 Lesson 2.6 Approximating Irrational Numbers** as a quick guide to resolve any of their doubts.

## Spectrum Math Grade 8 Chapter 2 Lesson 2.6 Approximating Irrational Numbers Answers Key

Approximate the value of an irrational number by exploring values.

The value of \(\sqrt{2}\) is something between 1 and 2.

Look at the squares of 1.4 and 1.5.

1.4^{2} = 1.96

1.5^{2} = 2.25

By looking at these squares, it is evident that \(\sqrt{2}\) is between 1.4 and 1.5.

**Approximate the value of each root to the tenths place.**

Question 1.

The value of \(\sqrt{7}\) is between ____ and _____

Answer: The value of \(\sqrt{7}\) is something between 2 and 3.

Look at the squares of 2.6 and 2.7.

2.6^{2} = 6.76

2.7^{2} = 7.29

By looking at these squares, it is evident that \(\sqrt{7}\) is between 2.6 and 2.7.

Question 2.

The value of \(\sqrt{10}\) is between ___ and _____

Answer: The value of \(\sqrt{10}\) is something between 3 and 4.

Look at the squares of 3.1 and 3.2.

3.1^{2} = 9.61

3.2^{2} = 10.24

By looking at these squares, it is evident that \(\sqrt{2}\) is between 3.1 and 3.2.

Question 3.

The value of \(\sqrt{26}\) is between ____ and _____

Answer: The value of \(\sqrt{26}\) is something between 5 and 6.

Look at the squares of 5 and 5.1.

5^{2} = 25

5.1^{2} = 26.01

By looking at these squares, it is evident that \(\sqrt{26}\) is between 5 and 5.1.

Question 4.

The value of \(\sqrt[3]{25}\) is between ____ and _____

Answer: The value of \(\sqrt[3]{25}\) is something between 2 and 3.

Look at the squares of 2.7 and 2.8.

2.7^{3} = 19.683

2.8^{3} = 21.952

By looking at these squares, it is evident that \(\sqrt[3]{25}\) is between 2.7 and 2.8.

Question 5.

The value of \(\sqrt[3]{99}\) is between ____ and ____

Answer: The value of \(\sqrt[3]{99}\) is something between 4 and 5.

Look at the squares of 4.6 and 4.7.

4.6^{3} = 97.336

4.7^{3} = 103.823

By looking at these squares, it is evident that \(\sqrt[3]{99}\) is between 4.6 and 4.7.

Question 6.

The value of \(\sqrt[3]{514}\) is between ___ and _____

Answer: The value of \(\sqrt[3]{514}\) is something between 8 and 9.

Look at the squares of 8 and 8.1.

8^{3} = 512

8.1^{3} = 531.441

By looking at these squares, it is evident that \(\sqrt[3]{514}\) is between 8 and 8.1.

Question 7.

The value of \(\sqrt{78}\) is between ____ and ____

Answer: The value of \(\sqrt{78}\) is something between 8 and 9.

Look at the squares of 8.8 and 8.9.

8.8^{2} = 77.44

8.9^{2} = 79.21

By looking at these squares, it is evident that \(\sqrt{78}\) is between 8.8 and 8.9.

Question 8.

The value of \(\sqrt[3]{824}\) is between ___ and ____

Answer: The value of \(\sqrt[3]{824}\) is something between 9 and 10.

Look at the squares of 9.3 and 9.4.

9.3^{3} = 804.357

9.4^{3} = 830.584

By looking at these squares, it is evident that \(\sqrt[3]{824}\) is between 9.3 and 9.4.

You can approximate the value of an irrational number to the hundredths place as well.

The value of \(\sqrt{3}\) is something between 1 and 2.

Look at the squares of 1.7 and 1.8.

1.7^{2} = 2.89

1.8^{2} = 3.24

By looking at these squares, it is evident that \(\sqrt{3}\) is between 1.7 and 1.8.

Now, narrow explore to the hundredths place.

1.73^{2}= 2.99

1.714^{2} = 3.03

By looking at these squares, it is evident that \(\sqrt{3}\) is between 1.73 and 1.74.

**Approximate each value to the hundredths place.**

Question 1.

The value of \(\sqrt{8}\) is between ___ and ____

Answer: The value of \(\sqrt{8}\) is something between 2 and 3.

Look at the squares of 2.8 and 2.9.

2.8^{2} = 7.84

2.9^{2} = 8.41

By looking at these squares, it is evident that \(\sqrt{8}\) is between 2.8 and 2.9.

Question 2.

The value of \(\sqrt{11}\) is between ____ and ____

Answer: The value of \(\sqrt{11}\) is something between 3 and 4.

Look at the squares of 3.3 and 3.4.

3.3^{2} = 10.89

3.4^{2} = 11.56

By looking at these squares, it is evident that \(\sqrt{11}\) is between 3.3 and 3.4.

Question 3.

The value of \(\sqrt{90}\) is between ____ and ____

Answer: The value of \(\sqrt{90}\) is something between 9 and 100.

Look at the squares of 9.4 and 9.5.

9.4^{2} = 88.36

9.5^{2} = 90.25

By looking at these squares, it is evident that \(\sqrt{90}\) is between 9.4 and 9.5.

Question 4.

The value of \(\sqrt[3]{72}\) is between ____ and ____

Answer: The value of \(\sqrt[3]{72}\) is something between 4 and 5.

Look at the squares of 4.1 and 4.2.

4.1^{3} = 68.921

4.2^{3} = 74.088

By looking at these squares, it is evident that \(\sqrt[3]{72}\) is between 4.1 and 4.2.

Question 5.

The value of \(\sqrt[3]{81}\) is between ____ and ____

Answer: The value of \(\sqrt[3]{81}\) is something between 4 and 5.

Look at the squares of 4.3 and 4.4.

4.3^{3} = 79.507

4.4^{3} = 85.184

By looking at these squares, it is evident that \(\sqrt[3]{81}\) is between 4.3 and 4.4.

Question 6.

The value of \(\sqrt{27}\) is between ____ and ____

Answer: The value of \(\sqrt{27}\) is something between 5 and 6.

Look at the squares of 5.1 and 5.2.

5.1^{2} = 26.01

5.2^{2} = 27.04

By looking at these squares, it is evident that \(\sqrt{27}\) is between 5.1 and 5.2.

Question 7.

The value of \(\sqrt[3]{33}\) is between ____ and ____

Answer: The value of \(\sqrt[3]{33}\) is something between 3 and 4.

Look at the squares of 3.2 and 3.3.

3.2^{3} = 32.768

3.3^{3} = 35.937

By looking at these squares, it is evident that \(\sqrt[3]{33}\) is between 3.2 and 3.3.

Question 8.

The value of \(\sqrt{23}\) is between ____ and _____

Answer: The value of \(\sqrt{23}\) is something between 4 and 5.

Look at the squares of 4.7 and 4.8.

4.7^{2} = 22.09

4.8^{2} = 23.09

By looking at these squares, it is evident that \(\sqrt{90}\) is between 4.7 and 4.8.