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

## Spectrum Math Grade 8 Chapter 2 Posttest Answers Key

**Rational and Irrational Number Relationships**

**Evaluate each expression.**

Question 1.

a. \(\sqrt{36}\) = _____

Answer:

The square root of a number is the number that, multiplied by itself, equals that number. The square root of 36 is 6.

\(\sqrt{36}\) = 6.

Numbers that have a whole number as their square root are called perfect squares. The expression of a square root is called a radical. The symbol is called a radical sign. When a number is not a perfect square, you can estimate its square root by determining which perfect squares it comes between.

b. \(\sqrt{16}\) = _____

Answer: 4

The square root of a number is the number that, multiplied by itself, equals that number. The square root of 16 is 4.

\(\sqrt{16}\) = 6.

Numbers that have a whole number as their square root are called perfect squares. The expression of a square root is called a radical. The symbol is called a radical sign. When a number is not a perfect square, you can estimate its square root by determining which perfect squares it comes between.

c. \(\sqrt{121}\) = _____

Answer: 11

The square root of a number is the number that, multiplied by itself, equals that number. The square root of 121 is 11.

\(\sqrt{121}\) = 11.

Numbers that have a whole number as their square root are called perfect squares. The expression of a square root is called a radical. The symbol is called a radical sign. When a number is not a perfect square, you can estimate its square root by determining which perfect squares it comes between.

Question 2.

a. \(\sqrt{\frac{9}{36}}\) = _____

Answer: \(\sqrt{\frac{9}{36}}\) = \(\frac{3}{6}\)

Numbers that have a whole number as their square root are called perfect squares. The expression of a square root is called a radical. The symbol is called a radical sign. When a number is not a perfect square, you can estimate its square root by determining which perfect squares it comes between.

b. \(\sqrt{144}\) = _____

Answer: 12

The square root of a number is the number that, multiplied by itself, equals that number. The square root of 144 is 12.

\(\sqrt{144}\) = 12.

Numbers that have a whole number as their square root are called perfect squares. The expression of a square root is called a radical. The symbol is called a radical sign. When a number is not a perfect square, you can estimate its square root by determining which perfect squares it comes between.

c. \(\sqrt{\frac{100}{121}}\) = ____

Answer: \(\sqrt{\frac{100}{121}}\) = \(\frac{10}{11}\)

Numbers that have a whole number as their square root are called perfect squares. The expression of a square root is called a radical. The symbol is called a radical sign. When a number is not a perfect square, you can estimate its square root by determining which perfect squares it comes between.

Question 3.

a. \(\sqrt[3]{512}\) = ____

Answer: 8

The cube of a number is that number multiplied by itself three times. A cube is expressed as n^{3}, which means n × n × n or n cubed. The cube root of a number is the number that, multiplied by itself and by itself again, equals that number. The cube root of 512 is 8.

= 8

The expression of a cube root is called a radical. The symbol is called a radical sign. The 3 on the radical sign shows that this is a cube root.

b. \(\sqrt[3]{1,000}\) = ____

Answer: 10

The cube of a number is that number multiplied by itself three times. A cube is expressed as n^{3}, which means n × n × n or n cubed. The cube root of a number is the number that, multiplied by itself and by itself again, equals that number. The cube root of 1000 is 10.

= 10

The expression of a cube root is called a radical. The symbol is called a radical sign. The 3 on the radical sign shows that this is a cube root.

c. \(\sqrt[3]{125}\) = ____

Answer: 5

The cube of a number is that number multiplied by itself three times. A cube is expressed as n^{3}, which means n × n × n or n cubed. The cube root of a number is the number that, multiplied by itself and by itself again, equals that number. The cube root of 125 is 5.

= 5

The expression of a cube root is called a radical. The symbol is called a radical sign. The 3 on the radical sign shows that this is a cube root.

Question 4.

a. \(\sqrt[3]{216}\) = ____

Answer: 6

The cube of a number is that number multiplied by itself three times. A cube is expressed as n^{3}, which means n × n × n or n cubed. The cube root of a number is the number that, multiplied by itself and by itself again, equals that number. The cube root of 216 is 6.

= 6

The expression of a cube root is called a radical. The symbol is called a radical sign. The 3 on the radical sign shows that this is a cube root.

b. \(3 \sqrt{\frac{64}{512}}\) = ____

Answer: \(\frac{4}{8}\)

Fractions can also have cube roots.

\(3 \sqrt{\frac{64}{512}}\) = \(\frac{4}{8}\)

because \(\frac{4}{8}\) × \(\frac{4}{8}\) × \(\frac{4}{8}\) = \(\frac{64}{512}\)

c. \(3 \sqrt{\frac{8}{729}}\) = ____

Answer: \(\frac{2}{9}\)

Fractions can also have cube roots.

\(3 \sqrt{\frac{8}{729}}\) = \(\frac{2}{9}\)

because \(\frac{2}{9}\) × \(\frac{2}{9}\) × \(\frac{2}{9}\) =\(\frac{8}{729}\)

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

Question 5.

The value of \(\sqrt{12}\) is between ___ and ____.

Answer: The value of \(\sqrt{12}\) is between 3.4 and 3.5.

The value of \(\sqrt{12}\) is between 3 and 4.

3.4^{2} = 11.56

3.5^{2} = 12.25

By looking at these squares, it is evident that The value of \(\sqrt{12}\) is between 3.4 and 3.5.

Question 6.

The value of \(\sqrt[3]{76}\) is between ___ and ____.

Answer: The value of \(\sqrt[3]{76}\) is between 4.2 and 4.3.

The value of \(\sqrt[3]{76}\) is between 4 and 5.

4.2^{3} = 74.088

4.3^{3} = 79.507

By looking at these squares, it is evident that The value of \(\sqrt[3]{76}\) is between 4.2 and 4.3.

Question 7.

The value of \(\sqrt{46}\) is between ___ and ____.

Answer: The value of \(\sqrt{46}\) is between 6 and 7.

The value of \(\sqrt{46}\) is between 6.7 and 6.8.

6.7^{2} = 44.89

6.8^{2} = 46.24

By looking at these squares, it is evident that The value of \(\sqrt{46}\) is between 6.7 and 6.8.

Question 8.

The value of \(\sqrt[3]{21}\) is between ____ and ___.

Answer: The value of \(\sqrt[3]{21}\) is between 2 and 3.

The value of \(\sqrt[3]{21}\) is between 2.7 and 2.8.

2.7^{3} = 19.683

2.8^{3} = 21.952

By looking at these squares, it is evident that The value of \(\sqrt[3]{21}\) is between 2.7 and 2.8

Question 9.

The value of \(\sqrt[3]{7}\) is between ___ and ____.

Answer: The value of \(\sqrt[3]{7}\) is between 1 and 2.

The value of \(\sqrt[3]{7}\) is between 1.9 and 2.

1.9^{3} = 6.859

2^{3} = 8

By looking at these squares, it is evident that The value of \(\sqrt[3]{7}\) is between 1.9 and 2.

Question 10.

The value of \(\sqrt{30}\) is between ____ and ____.

Answer: The value of \(\sqrt{30}\) is between 5 and 6.

The value of \(\sqrt{30}\) is between 5.4 and 5.5.

5.4^{2} = 29.16

5.5^{2} = 30.25

By looking at these squares, it is evident that The value of \(\sqrt{30}\) is between 5.4 and 5.5.

**Use roots or exponents to solve each equation. Write fractions in simplest form.**

Question 11.

a. \(\sqrt{x}\) = 7

x = ____

Answer: x = 49

\(\sqrt{x}\)= 7

As the exponent is 2, so use the square root as the inverse operation.

Square both sides of the equation.

{\(\sqrt{x}\)}^{2 }= {7}^{2}

By simplification,

x = 49

b. x^{3} = 512

x = _______

Answer: x = 8

x^{3} = 512

As the exponent is 3, so use the cube root as the inverse operation.

Use root on both sides

\(\sqrt[3]{x^{3}}\) = \(\sqrt[3]{512}\)

By simplification,

x = 8

c. x^{2} = 81

x = _______

Answer: x = 9

x^{2} = 81

As the exponent is 2, so use the square root as the inverse operation.

Use root on both sides

\(\sqrt{x^{2} }\) = \(\sqrt{81}\)

By simplification,

x = 9

Question 12.

a. x^{2} = 144

x = _______

Answer: x = 12

x^{2} = 144

As the exponent is 2, so use the square root as the inverse operation.

Use root on both sides

\(\sqrt{x^{2} }\) = \(\sqrt{144}\)

By simplification,

x = 12

b. \(\sqrt[3]{x}\) = 4

x = _______

Answer: x = 64

\(\sqrt[3]{x}\) = 4

As the exponent is 3, so use the cube root as the inverse operation.

Square both sides of the equation.

{\(\sqrt[3]{x}\)}^{3 }= {4}^{3}

By simplification,

x = 64

c. \(\sqrt[3]{x}\) = 9

x = _______

Answer: x = 729

\(\sqrt[3]{x}\) = 9

As the exponent is 3, so use the cube root as the inverse operation.

Square both sides of the equation.

{\(\sqrt[3]{x}\)}^{3 }= {9}^{3}

By simplification,

x = 729

**Compare using <, >, or =.**

Question 13.

a. \(\sqrt{\frac{9}{10}}\) _____ \(\frac{3}{4}\)

Answer: \(\sqrt{\frac{9}{10}}\) > \(\frac{3}{4}\)

This statement is true because \(\sqrt{\frac{9}{10}}\) is 0.9 and \(\frac{3}{4}\) is 0.7 . As 0.9 is greater than 0.7. Therefore, \(\sqrt{\frac{9}{10}}\) > \(\frac{3}{4}\).

b. \(\sqrt{12}\) _____ 3

Answer: \(\sqrt{12}\) > 3

This statement is true because \(\sqrt{12}\) is 3.46. As \(\sqrt{12}\)is greater than 3. Therefore, 3.46 is greater than \(\sqrt{12}\).

c. \(\sqrt[3]{27}\) _____ 3

Answer: \(\sqrt[3]{27}\) = 3

This statement is true because \(\sqrt[3]{27}\) is 3. Therefore, \(\sqrt[3]{27}\) is equal to 3.

Question 14.

a. 2.1 ____ \(\sqrt{4}\)

Answer: 2.1 > \(\sqrt{4}\)

This statement is true because \(\sqrt{4}\) is 2. As 2.1 is greater than 2. Therefore, 2.1 is greater than \(\sqrt{4}\).

b. \(\sqrt[3]{76}\) _____ 5.5

Answer: \(\sqrt[3]{76}\) < 5.5

This statement is true because \(\sqrt[3]{76}\) is 4.23. As 4.23is less than 5.5. Therefore, \(\sqrt[3]{76}\) < 5.5

c. \(\sqrt[3]{48}\) _____ 5.5

Answer: \(\sqrt[3]{48}\) < 5.5

This statement is true because \(\sqrt[3]{48}\) is 3.63. As 3.63 is less than 5.5. Therefore, \(\sqrt[3]{48}\) < 5.5

Question 15.

a. \(0 . \overline{66}\) _____ \(\sqrt[3]{\frac{8}{27}}\)

Answer: \(0 . \overline{66}\) = \(\sqrt[3]{\frac{8}{27}}\)

This statement is true because \(\sqrt[3]{\frac{8}{27}}\) is \(0 . \overline{66}\) . Therefore, \(0 . \overline{66}\) is equal to \(\sqrt[3]{\frac{8}{27}}\).

b. \(\frac{6}{7}\) _____ \(\sqrt{3}\)

Answer: \(\frac{6}{7}\) < \(\sqrt{3}\)

This statement is true because \(\frac{6}{7}\) is 0.85 and \(\sqrt{3}\) is 1.732 . As 0.85 is less than 1.732. Therefore, \(\frac{6}{7}\) < \(\sqrt{3}\)

c. \(\sqrt{7}\) ______ 3

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

This statement is true because \(\sqrt{7}\) is 2.64. As 2.64 is less than 3. Therefore, \(\sqrt{7}\) is less than 3.

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

Question 16.

2π, \(\sqrt{38}\), \(\sqrt{52}\)

Answer:

2π = 6.28

\(\sqrt{38}\) = 6.16

\(\sqrt{52}\) = 7.2

Question 17.

2.75, \(\sqrt{18}\), \(\sqrt[3]{27}\)

Answer:

\(\sqrt{18}\) = 4.24

\(\sqrt[3]{27}\) = 3

Question 18.

\(\sqrt{3}\), 1.4, \(\frac{3}{2}\)

Answer:

\(\sqrt{3}\) = 1.73

\(\frac{3}{2}\) = 1.5