Spectrum Math Grade 8 Chapter 2 Posttest Answer Key

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 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. \(\sqrt[3]{1,000}\) = ____
Answer: 10
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. \(\sqrt[3]{125}\) = ____
Answer: 5
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. \(\sqrt[3]{216}\) = ____
Answer: 6
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 = ____
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. x³ = 512
x = _______
Answer: x  = 8
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 = _______
Answer: x = 9
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 = _______
Answer: x = 12
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 = _______
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