Students can use the **Spectrum Math Grade 8 Answer Key** **Chapter 1 Posttest** as a quick guide to resolve any of their doubts.

## Spectrum Math Grade 8 Chapter 1 Posttest Answers Key

**Check What You Learned**

**Integers and Exponents**

**Find the value of each expression.**

Question 1.

a. 3^{7} = ____

Answer: 2187

A power of a number represents repeated multiplication of the number by itself.

3^{7} = 3 x 3 x 3 x 3 x 3 x 3 x 3 and is read 3 to the seventh power.

In exponential numbers, the base is the number that is multiplied, and the exponent represents the number of times the base is used as factor. In 3^{7}, 3 is the base and 7 is the exponent.

3^{7} means 3 is used as a factor 7 times.

3 x 3 x 3 x 3 x 3 x 3 x 3 = 3^{7}

3 x 3 x 3 x 3 x 3 x 3 x 3 = 2187

3^{7} = 2187

b. 4^{8} = ____

Answer: 65536

A power of a number represents repeated multiplication of the number by itself.

4^{8} = 4 x 4 x 4 x 4 x 4 x 4 x 4 x 4 and is read 4 to the eighth power.

In exponential numbers, the base is the number that is multiplied, and the exponent represents the number of times the base is used as factor. In 4^{8}, 4 is the base and 8 is the exponent.

4^{8} means 4 is used as a factor 8 times.

4 x 4 x 4 x 4 x 4 x 4 x 4 x 4 = 4^{8}

4 x 4 x 4 x 4 x 4 x 4 x 4 x 4 = 65536

4^{8} = 65536

c. 5^{2} = ____

Answer: 25

A power of a number represents repeated multiplication of the number by itself.

5² = 5 x 5 and is read 5 to the second power.

In exponential numbers, the base is the number that is multiplied, and the exponent represents the number of times the base is used as factor. In 5², 5 is the base and 2 is the exponent.

5² means 5 is used as a factor 2 times.

5 x 5 = 5²

5 x 5 = 25

5² = 25

Question 2.

a. 12^{9} = ____

Answer: 5159780352

A power of a number represents repeated multiplication of the number by itself.

12^{9} = 12 x 12 x 12 x 12 x 12 x 12 x 12 x 12 x 12 and is read 12 to the ninth power.

In exponential numbers, the base is the number that is multiplied, and the exponent represents the number of times the base is used as factor. In 12^{9}, 12 is the base and 9 is the exponent.

12^{9} means 12 is used as a factor 9 times.

12 x 12 x 12 x 12 x 12 x 12 x 12 x 12 x 12 = 12^{9}

12 x 12 x 12 x 12 x 12 x 12 x 12 x 12 x 12 = 5159780352

12^{5} = 5159780352

b. 4^{5} = ____

Answer: 1024

A power of a number represents repeated multiplication of the number by itself.

4^{5} = 4 × 4 × 4 × 4 × 4 and is read 4 to the fifth power.

In exponential numbers, the base is the number that is multiplied, and the exponent represents the number of times the base is used as factor. In 4^{5}, 4 is the base and 5 is the exponent.

4^{5} means 4 is used as a factor 5 times.

4 × 4 × 4 × 4 × 4 = 4^{5}

4 × 4 × 4 × 4 × 4 = 1024

4^{5} = 1024

c. 8^{4} = ____

Answer: 4096

A power of a number represents repeated multiplication of the number by itself.

8^{4} = 8 x 8 x 8 x 8 and is read 8 to the fourth power.

In exponential numbers, the base is the number that is multiplied, and the exponent represents the number of times the base is used as factor. In 8^{4}, 8 is the base and 4 is the exponent.

8^{4} means 8 is used as a factor 4 times.

8 x 8 x 8 x 8 = 8^{4}

8 x 8 x 8 x 8 = 4096

8^{4} = 4096

Question 3.

a. 3^{-6} = ____

Answer: 0.00137

When a power includes a negative exponent, express the number as 1 divided by the base, and change the exponent to positive.

3^{-6} = \(\frac{1}{3^{6}}\)

= \(\frac{1}{729}\)

= 0.00137

b. 4^{-3} = ____

Answer: 0.015625

When a power includes a negative exponent, express the number as 1 divided by the base, and change the exponent to positive.

4^{-3} = \(\frac{1}{4^{3}}\)

= \(\frac{1}{64}\)

= 0.015625

c. 5^{-7} = ____

Answer: 0.0000128

When a power includes a negative exponent, express the number as 1 divided by the base, and change the exponent to positive.

5^{-7} = \(\frac{1}{5^{7}}\)

= \(\frac{1}{78125}\)

= 0.0000128

Question 4.

a. 10^{-4} = ____

Answer: 0.0001

When a power includes a negative exponent, express the number as 1 divided by the base, and change the exponent to positive.

10^{-4} = \(\frac{1}{10^{4}}\)

= \(\frac{1}{10000}\)

= 0.0001

b. 6^{-3} = ____

Answer: 0.004629

When a power includes a negative exponent, express the number as 1 divided by the base, and change the exponent to positive.

6^{-3}= \(\frac{1}{6^{3}}\)

= \(\frac{1}{216}\)

= 0.004629

c. 8^{-5} = ____

Answer: 0.0000305

When a power includes a negative exponent, express the number as 1 divided by the base, and change the exponent to positive.

8^{-5}= \(\frac{1}{8^{5}}\)

= \(\frac{1}{32768}\)

= 0.0000305

Question 5.

a. 8^{-6} = ____

Answer: 0.000003814

When a power includes a negative exponent, express the number as 1 divided by the base, and change the exponent to positive.

8^{-6} = \(\frac{1}{8^{6}}\)

= \(\frac{1}{262144}\)

= 0.000003814

b. 7^{4} = ____

Answer: 2401

A power of a number represents repeated multiplication of the number by itself.

7^{4} = 7 x 7 x 7 x 7 and is read 7 to the fourth power.

In exponential numbers, the base is the number that is multiplied, and the exponent represents the number of times the base is used as factor. In 7^{4}, 7 is the base and 4 is the exponent.

7^{4} means 7 is used as a factor 4 times.

7 x 7 x 7 x 7 = 7^{4}

7 x 7 x 7 x 7 = 2401

7^{4} = 2401

c. 3^{-9} = ____

Answer: 0.00137

When a power includes a negative exponent, express the number as 1 divided by the base, and change the exponent to positive.

3^{-9} = \(\frac{1}{3^{9}}\)

= \(\frac{1}{729}\)

= 0.00137

Question 6.

a. 10^{7} = ____

Answer: 10000000

A power of a number represents repeated multiplication of the number by itself.

10^{7} = 10 x 10 x 10 x 10 x 10 x 10 x 10 and is read 10 to the seventh power.

In exponential numbers, the base is the number that is multiplied, and the exponent represents the number of times the base is used as factor. In 10^{7}, 10 is the base and 7 is the exponent.

10^{7} means 10 is used as a factor 7 times.

10 x 10 x 10 x 10 x 10 x 10 x 10 = 10^{7}

10 x 10 x 10 x 10 x 10 x 10 x 10 = 10000000

3^{7} = 10000000

b. 9^{-2} = ____

Answer: 0.012345

When a power includes a negative exponent, express the number as 1 divided by the base, and change the exponent to positive.

9^{-2}= \(\frac{1}{9^{2}}\)

= \(\frac{1}{81}\)

= 0.012345

c. 2^{8} = ____

Answer: 256

A power of a number represents repeated multiplication of the number by itself.

2^{8} = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 and is read 2 to the eighth power.

In exponential numbers, the base is the number that is multiplied, and the exponent represents the number of times the base is used as factor. In 2^{8}, 2 is the base and 8 is the exponent.

2^{8} means 2 is used as a factor 8 times.

2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 = 2^{8}

2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 = 256

2^{8} = 256

**Rewrite each multiplication or division expression using a base and an exponent.**

Question 7.

a. 8^{2} × 8^{3} = _____

Answer: 32768

To multiply powers with the same base, combine bases, add the exponents, then simplify.

8^{2} × 8^{3} = 8^{2 + 3} = 8^{5} = 32768

b. 5^{-5} × 5^{-2} = _____

Answer: 0.0000128

To multiply or divide powers with the same base, combine bases, add or subtract the exponents.

For multiplication, add the exponents by combining the bases.

5^{-5} × 5^{-2} = 5^{-7
}By simplification,

= \(\frac{1}{5^{7}}\)

= \(\frac{1}{78125}\)

= 0.0000128

c. 6^{2} × 6^{4} = _____

Answer: 46656

To multiply powers with the same base, combine bases, add the exponents, then simplify.

6^{2} × 6^{4} = 6^{2 + 4} = 6^{6} = 46656

Question 8.

a. 4^{-1} × 4^{3} = _____

Answer: 16

To multiply or divide powers with the same base, combine bases, add or subtract the exponents.

For multiplication, add the exponents by combining the bases.

4^{-1} × 4^{3} = 4^{2
}By simplification,

= 16

b. 3^{4} ÷ 3^{-3} = ____

Answer: 2187

To multiply or divide powers with the same base, combine bases, add or subtract the exponents.

For division, subtract the exponents by combining the bases.

3^{4} ÷ 3^{-3} = 3^{7}^{
}By simplification,

= 2187

c. 12^{-2} ÷ 12^{4} = ____

Answer: 0.000000334

To multiply or divide powers with the same base, combine bases, add or subtract the exponents.

For division, subtract the exponents by combining the bases.

12^{-2} ÷ 12^{4} = 12^{-6}^{
}By simplification,

= \(\frac{1}{12^{6}}\)

= \(\frac{1}{2985984}\)

= 0.000000334

Question 9.

a. 5^{4} × 5^{7} = ____

Answer: 48828125

To multiply powers with the same base, combine bases, add the exponents, then simplify.

5^{4} × 5^{7} = 5^{4 + 7} = 5^{11} = 48828125

b. 8^{-2} × 8^{-6} = ____

Answer: 0.0000000596

To multiply or divide powers with the same base, combine bases, add or subtract the exponents.

For multiplication, add the exponents by combining the bases.

8^{-2} × 8^{-6}= 8^{-8
}By simplification,

= \(\frac{1}{5^{7}}\)

= \(\frac{1}{16777216}\)

= 0.0000000596

c. 5^{8} × 5^{-3} = ____

Answer: 0.00032

To multiply or divide powers with the same base, combine bases, add or subtract the exponents.

For multiplication, add the exponents by combining the bases.

5^{8} × 5^{-3} = 5^{5
}By simplification,

= \(\frac{1}{5^{7}}\)

= \(\frac{1}{3125}\)

= 0.00032

Question 10.

a. 9^{-2} × 9^{-5} = ____

Answer: 0.000000209

To multiply or divide powers with the same base, combine bases, add or subtract the exponents.

For multiplication, add the exponents by combining the bases.

9^{-2} × 9^{-5} = 9^{-7
}By simplification,

= \(\frac{1}{9^{7}}\)

= \(\frac{1}{4782969}\)

= 0.000000209

b. 7^{8} ÷ 7^{-3} = ____

Answer: 1977326743

To multiply or divide powers with the same base, combine bases, add or subtract the exponents.

For division, subtract the exponents by combining the bases.

7^{8} ÷ 7^{-3}= 7^{11}^{
}By simplification,

= 1977326743

c. 6^{-2} ÷ 6^{-4} = ____

Answer: 36

To multiply or divide powers with the same base, combine bases, add or subtract the exponents.

For division, subtract the exponents by combining the bases.

6^{-2} ÷ 6^{-4} = 6^{2}

By simplification,

= 36

Question 11.

a. 7^{-1} × 7^{-3} = ____

Answer: 0.0004164

To multiply or divide powers with the same base, combine bases, add or subtract the exponents.

For multiplication, add the exponents by combining the bases.

7^{-1} × 7^{-3}= 7^{-4
}By simplification,

= \(\frac{1}{7^{4}}\)

= \(\frac{1}{2401}\)

= 0.0004164

b. 9^{4} ÷ 9^{8} = ____

Answer: 282429536481

To multiply or divide powers with the same base, combine bases, add or subtract the exponents.

For division, subtract the exponents by combining the bases.

9^{4} ÷ 9^{8} = 9^{12}^{
}By simplification,

= 282429536481

c. 3^{-8} ÷ 3^{4} = ____

Answer: 0.00000188

To multiply or divide powers with the same base, combine bases, add or subtract the exponents.

For division, subtract the exponents by combining the bases.

3^{-8} ÷ 3^{4} = 3^{-12}^{
}By simplification,

= \(\frac{1}{3^{10}}\)

= \(\frac{1}{531441}\)

= 0.00000188

Question 12.

a. 10^{-3} × 10^{3} = ____

Answer: 1

To multiply or divide powers with the same base, combine bases, add or subtract the exponents.

For multiplication, add the exponents by combining the bases.

10^{-3} × 10^{3} = 10^{0
}By simplification,

= 1

b. 8^{6} ÷ 8^{-3} = ____

Answer: 134217728

To multiply or divide powers with the same base, combine bases, add or subtract the exponents.

For division, subtract the exponents by combining the bases.

8^{6} ÷ 8^{-3} = 8^{9}^{
}By simplification,

= 134217728

c. 7^{4} × 7^{2} = ____

Answer: 117649

To multiply powers with the same base, combine bases, add the exponents, then simplify.

7^{4} × 7^{2} = 7^{4 + 2} = 7^{6} = 117649

**Rewrite each number in standard notation.**

Question 13.

a. 3.04 × 10^{-3}

__________________

Answer: 0.00304

Translate numbers written in scientific notation into standard form by reading the exponent.

Move the decimal left 3 times

3.04 × 10^{-3} = 0.00304

b. 4.26 × 10^{2}

__________________

Answer: 426

Translate numbers written in scientific notation into standard form by reading the exponent.

Move the decimal right 2 times

4.26 × 10^{2}= 426

c. 8.1 × 10^{-4}

__________________

Answer: 0.00081

Translate numbers written in scientific notation into standard form by reading the exponent.

Move the decimal left 4 times

8.1 × 10^{-4}= 0.00081

Question 14.

a. 6.5 × 10^{4}

__________________

Answer: 65000

Translate numbers written in scientific notation into standard form by reading the exponent.

Move the decimal right 4 times

6.5 × 10^{4}= 65000

b. 2.4 × 10^{-2}

__________________

Answer: 0.024

Translate numbers written in scientific notation into standard form by reading the exponent.

Move the decimal left 2 times

2.4 × 10^{-2} = 0.024

c. 7.15 × 10

__________________

Answer: 71.5

Translate numbers written in scientific notation into standard form by reading the exponent.

Move the decimal right 1 times

7.15 × 10= 71.5

Question 15.

a. 3.286 × 10^{-5}

__________________

Answer: 0.00003286

Translate numbers written in scientific notation into standard form by reading the exponent.

Move the decimal left 5 times

3.286 × 10^{-5} = 0.00003286

b. 8.2734 × 10^{6}

__________________

Answer: 8273400

Translate numbers written in scientific notation into standard form by reading the exponent.

Move the decimal right 6 times

8.2734 × 10^{6}= 8273400

c. 7.362 × 10^{-6}

__________________

Answer: 0.000007362

Translate numbers written in scientific notation into standard form by reading the exponent.

Move the decimal left 6 times

7.362 × 10^{-6} = 0.000007362

Question 16.

a. 8.23 × 10^{7}

__________________

Answer: 82300000

Translate numbers written in scientific notation into standard form by reading the exponent.

Move the decimal right 7 times

8.23 × 10^{7}= 82300000

b. 4.602 × 10^{-3}

__________________

Answer: 0.004602

Translate numbers written in scientific notation into standard form by reading the exponent.

Move the decimal left 3 times

4.602 × 10^{-3} = 0.004602

c. 2.382 × 10^{5}

__________________

Answer:238200

Translate numbers written in scientific notation into standard form by reading the exponent.

Move the decimal right 5 times

2.382 × 10^{5}= 238200

Question 17.

a. 9.12 × 10^{7}

__________________

Answer: 91200000

Translate numbers written in scientific notation into standard form by reading the exponent.

Move the decimal right 7 times

9.12 × 10^{7}= 91200000

b. 7.292 × 10^{-3}

__________________

Answer: 0.007292

Translate numbers written in scientific notation into standard form by reading the exponent.

Move the decimal left 3 times

7.292 × 10^{-3} = 0.007292

c. 8.153 × 10^{-4}

__________________

Answer: 0.0008153

Translate numbers written in scientific notation into standard form by reading the exponent.

Move the decimal left 4 times

8.153 × 10^{-4 }= 0.0008153

**Rewrite each number in scientific notation.**

Question 18.

a. 1,985

__________________

Answer: 1.985× 10^{3}

Scientific notation is most often used as a concise way of writing very large and very small numbers. It is written as a number between 1 and 10 multiplied by a power of 10.

Move the decimal left 3 places and multiply the shift by the power of 10.

1,985 =1.985× 10^{3}

b. 10.32

__________________

Answer: 1.032× 10^{1}

Scientific notation is most often used as a concise way of writing very large and very small numbers. It is written as a number between 1 and 10 multiplied by a power of 10.

Move the decimal left 1 places and multiply the shift by the power of 10.

10.32 =1.032× 10^{1}

c. 414.1

__________________

Answer: 4.141× 10^{2}

Scientific notation is most often used as a concise way of writing very large and very small numbers. It is written as a number between 1 and 10 multiplied by a power of 10.

Move the decimal left 2 places and multiply the shift by the power of 10.

414.1 =4.141× 10^{2}

Question 19.

a. 0.0003954

__________________

Answer: 3.954× 10^{-4}

Scientific notation is most often used as a concise way of writing very large and very small numbers. It is written as a number between 1 and 10 multiplied by a power of 10.

Move the decimal right 4 places and multiply the shift by the power of 10.

0.0003954 = 3.954× 10^{-4}

b. 95.45

__________________

Answer: 9.545× 10^{1}

Scientific notation is most often used as a concise way of writing very large and very small numbers. It is written as a number between 1 and 10 multiplied by a power of 10.

Move the decimal left 1 places and multiply the shift by the power of 10.

95.45 =9.545× 10^{1}

c. 8,524

__________________

Answer: 8.524× 10^{3}

Scientific notation is most often used as a concise way of writing very large and very small numbers. It is written as a number between 1 and 10 multiplied by a power of 10.

Move the decimal left 3 places and multiply the shift by the power of 10.

8,524 = 8.524× 10^{3}

Question 20.

a. 239,390

__________________

Answer: 2.39390× 10^{5}

Scientific notation is most often used as a concise way of writing very large and very small numbers. It is written as a number between 1 and 10 multiplied by a power of 10.

Move the decimal left 5 places and multiply the shift by the power of 10.

239,390 = 2.39390× 10^{5}

b. 0.003121

__________________

Answer: 3.121× 10^{-3}

Scientific notation is most often used as a concise way of writing very large and very small numbers. It is written as a number between 1 and 10 multiplied by a power of 10.

Move the decimal right 3 places and multiply the shift by the power of 10.

0.003121 = 3.121× 10^{-3}

c. 43,100

__________________

Answer: 4.3100× 10^{4}

Scientific notation is most often used as a concise way of writing very large and very small numbers. It is written as a number between 1 and 10 multiplied by a power of 10.

Move the decimal left 4 places and multiply the shift by the power of 10.

43,100 = 4.3100× 10^{4}

Question 21.

a. 0.0283

__________________

Answer: 2.83× 10^{-2}

Scientific notation is most often used as a concise way of writing very large and very small numbers. It is written as a number between 1 and 10 multiplied by a power of 10.

Move the decimal right 2 places and multiply the shift by the power of 10.

0.0283 = 2.83× 10^{-2}

b. 0.000273

__________________

Answer: 2.73× 10^{-4}

Scientific notation is most often used as a concise way of writing very large and very small numbers. It is written as a number between 1 and 10 multiplied by a power of 10.

Move the decimal right 4 places and multiply the shift by the power of 10.

0.000273= 2.73× 10^{-4}

c. 3,476,000

__________________

Answer: 3.476000×10^{6}

Scientific notation is most often used as a concise way of writing very large and very small numbers. It is written as a number between 1 and 10 multiplied by a power of 10.

Move the decimal left 6 places and multiply the shift by the power of 10.

3,476,000 = 3.476000×10^{6}

Question 22.

a. 37, 120,000

__________________

Answer: 3.7120000× 10^{7}

Scientific notation is most often used as a concise way of writing very large and very small numbers. It is written as a number between 1 and 10 multiplied by a power of 10.

Move the decimal left 7 places and multiply the shift by the power of 10.

37, 120,000 = 3.7120000× 10^{7}

b. 374,200

__________________

Answer: 3.74200× 10^{5}

Scientific notation is most often used as a concise way of writing very large and very small numbers. It is written as a number between 1 and 10 multiplied by a power of 10.

Move the decimal left 5 places and multiply the shift by the power of 10.

374,200 = 3.74200× 10^{5}

c. 0.005283

__________________

Answer: 5.283× 10^{-3}

Scientific notation is most often used as a concise way of writing very large and very small numbers. It is written as a number between 1 and 10 multiplied by a power of 10.

Move the decimal right 3 places and multiply the shift by the power of 10.

0.005283 = 5.283× 10^{-3}