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

Rational and Irrational Number Relationships

Evaluate each expression.

Question 1.
a. $$\sqrt{36}$$ = _____
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. $$\sqrt{16}$$ = _____
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. $$\sqrt{121}$$ = _____
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. $$\sqrt{144}$$ = _____
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. $$\sqrt[3]{512}$$ = ____
The cube of a number is that number multiplied by itself three times. A cube is expressed as n3, 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. $$\sqrt[3]{1,000}$$ = ____
The cube of a number is that number multiplied by itself three times. A cube is expressed as n3, 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. $$\sqrt[3]{125}$$ = ____
The cube of a number is that number multiplied by itself three times. A cube is expressed as n3, 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. $$\sqrt[3]{216}$$ = ____
The cube of a number is that number multiplied by itself three times. A cube is expressed as n3, 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.
3.42 = 11.56
3.52 = 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.
4.23 = 74.088
4.33 = 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.
6.72 = 44.89
6.82 = 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.
2.73 = 19.683
2.83 = 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.
1.93 = 6.859
23 = 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.
5.42 = 29.16
5.52 = 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 = ____
$$\sqrt{x}$$= 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. x3 = 512
x = _______
x3 = 512
As the exponent is 3, so use the cube root as the inverse operation.
Use root on both sides
$$\sqrt[3]{x3}$$ = $$\sqrt[3]{512}$$
By simplification,
x  = 8

c. x2 = 81
x = _______
x2 = 81
As the exponent is 2, so use the square root as the inverse operation.
Use root on both sides
$$\sqrt{x2 }$$ = $$\sqrt{81}$$
By simplification,
x = 9

Question 12.
a. x2 = 144
x = _______
x2 = 144
As the exponent is 2, so use the square root as the inverse operation.
Use root on both sides
$$\sqrt{x2 }$$ = $$\sqrt{144}$$
By simplification,
x = 12

b. $$\sqrt[3]{x}$$ = 4
x = _______
$$\sqrt[3]{x}$$ = 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 = _______
$$\sqrt[3]{x}$$ = 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, $$\sqrt{38}$$, $$\sqrt{52}$$

2π = 6.28
$$\sqrt{38}$$ = 6.16
$$\sqrt{52}$$ = 7.2

Question 17.
2.75, $$\sqrt{18}$$, $$\sqrt[3]{27}$$

$$\sqrt{18}$$ = 4.24
$$\sqrt[3]{27}$$ = 3
$$\sqrt{3}$$, 1.4, $$\frac{3}{2}$$
$$\sqrt{3}$$ = 1.73
$$\frac{3}{2}$$ = 1.5