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

## Spectrum Math Grade 8 Chapter 2 Pretest Answers Key

**Check What You Know**

**Rational and Irrational Number Relationships**

**Evaluate each expression.**

Question 1.

a.

= ____

Answer: 5

The square root of a number is the number that, multiplied by itself, equals that number. The square root of 25 is 5.

= 5.

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.

= ____

Answer: 3

The square root of a number is the number that, multiplied by itself, equals that number. The square root of 9 is 3. = 3.

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.

Answer: 10

The square root of a number is the number that, multiplied by itself, equals that number. The square root of 100 is 10.

= 10.

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.

= ____

Answer: \(\frac{2}{4}\)

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

= \(\frac{2}{4}\)

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. = ____

Answer: 9

The square root of a number is the number that, multiplied by itself, equals that number. The square root of 81 is 9.

= 9.

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. = ____

Answer: \(\frac{3}{5}\)

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

= \(\frac{3}{5}\)

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.

Answer: 7

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 343 is 7.

= 7

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.

Answer: 9

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 729 is 9.

= 9

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.

Answer: 4

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 64 is 4.

= 4

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.

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.

Answer: \(\frac{3}{8}\)

Fractions can also have cube roots.

= \(\frac{3}{8}\)

because \(\frac{3}{8}\) x \(\frac{3}{8}\) x \(\frac{3}{8}\) = \(\frac{27}{512}\)

c.

Answer: \(\frac{4}{9}\)

Fractions can also have cube roots.

= \(\frac{4}{9}\)

because \(\frac{4}{9}\) x \(\frac{4}{9}\) x \(\frac{4}{9}\) = \(\frac{64}{729}\)

**Approximate the value of each expression.**

Question 5.

The value of is between ___ and ____

Answer: The value of 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 is between 3.1 and 3.2.

Question 6.

The value of is between ___ and ____

Answer: The value of 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 is between 4.1 and 4.2.

Question 7.

The value of is between ___ and ____

Answer: The value of is something between 6 and 7.

Look at the squares of 6.5 and 6.6.

6.5^{2} = 42.25

6.6^{2} = 43.56

By looking at these squares, it is evident that is between 6.5 and 6.6.

Question 8.

The value of is between ___ and ____

Answer: The value of is something between 2 and 3.

Look at the squares of 2.5 and 2.6.

2.5^{3} = 15.625

2.6^{3} = 17.576

By looking at these squares, it is evident that is between 2.5 and 2.6.

Question 9.

The value of is between ___ and ____

Answer: The value of is something between 1 and 2.

Look at the squares of 1.2 and 1.3.

1.2^{3} = 1.728

1.3^{3} = 2.197

By looking at these squares, it is evident that is between 1.2 and 1.3.

Question 10.

The value of is between ___ and ____

Answer: The value of is something between 4 and 5.

Look at the squares of 4.8 and 4.9.

4.8^{2} = 23.04

4.9^{2} = 24.01

By looking at these squares, it is evident that is between 4.8 and 4.9.

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

Question 11.

a. x^{2} = 64

x = ___

Answer: x = 8

x^{2} = 64

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

Use root on both sides

\(\sqrt{x^{2} }\) =\(\sqrt{64}\)

By simplification,

x = 8

b. = 9

x = ____

Answer: x = 81

\(\sqrt{x}\)= 81

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

Square both sides of the equation.

{\(\sqrt{x}\)}^{2 }= {9}^{2}

By simplification,

x = 81

c. x^{3} = 343

x = ___

Answer: x = 7

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

Use root on both sides

\(\sqrt[3]{x^{3}}\) = \(\sqrt[3]{343}\)

By simplification,

x = 7

Question 12.

a.

= 6

x = ____

Answer: x = 216

\(\sqrt[3]{x}\) = 6

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

Square both sides of the equation.

{\(\sqrt[3]{x}\)}^{3 }= {6}^{3}

By simplification,

x = 216

b.

x^{2} = 121

x = ____

Answer: x = 11

x^{2} = 121

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

Use root on both sides

\(\sqrt{x^{2} }\) =\(\sqrt{121}\)

By simplification,

x = 11

c. = 10

x = ____

Answer: x = 1000

\(\sqrt[3]{x}\) = 10

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

Square both sides of the equation.

{\(\sqrt[3]{x}\)}^{3 }= {10}^{3}

By simplification,

x = 1000

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

Question 13.

a. _____ \(\frac{2}{3}\)

Answer: = \(\frac{2}{3}\)

This statement is true because is \(\frac{2}{3}\). Therefore, is equal to \(\frac{2}{3}\).

b. ____ 5

Answer: < 5

This statement is true because is 3.16. As 3.16 is less than 5. Therefore, is less than 5.

c. ____ 3

Answer: < 3

This statement is true because is 2.92. As 2.92 is less than 3. Therefore, is less than 3.

Question 14.

a. 1.2 ____

Answer: 1.2 <

This statement is true because is 2. As 1.2 is less than 2. Therefore, 1.2 is less than .

b. ___ 3.5

Answer: > 3.5

This statement is true because is 3.9. As 3.9 is greater than 3.5. Therefore, is greater than 3.5.

c. ____ 4

Answer: < 4

This statement is true because is 3.33. As 3.33 is less than 4. Therefore, is less than 4 .

Question 15.

a. \(0 . \overline{33}\) _____

Answer: \(0 . \overline{33}\) <

This statement is true because is 0.57. As \(0 . \overline{33}\) is less than 0.57. Therefore, \(0 . \overline{33}\) is less than .

b. \(\frac{5}{6}\) ____

Answer: \(\frac{5}{6}\) <

This statement is true because is 1.414 and \(\frac{5}{6}\) is 0.6. As 0.6 is less than 1.414. Therefore, \(\frac{5}{6}\) is less than .

c. ____ 3

Answer: < 3

This statement is true because is 2.23. As 2.23 is less than 3 Therefore, is less than 3.

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

Question 16.

14, , 4π

Answer:

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

Question 17.

Answer:

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

Question 18.

, 2, 5

Answer:

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