Students can use the Spectrum Math Grade 8 Answer Key Chapter 1 Lesson 1.3 Negative Exponents as a quick guide to resolve any of their doubts.
Spectrum Math Grade 8 Chapter 1 Lesson 1.3 Negative Exponents Answers Key
When a power includes a negative exponent, express the number as 1 divided by the base, and change the exponent to positive.
4-2 = \(\frac{1}{4^{2}}\)
= \(\frac{1}{16}\)
= 0.0625
To multiply or divide powers with the same base, combine bases, add or subtract the exponents, and then simplify.
2-3 × 2-2 = 2-5 = \(\frac{1}{2^{5}}\) = 0.03125
2-4 ÷ 2-2 = 2-2 = \(\frac{1}{2^{2}}\) = 0.25
Rewrite each expression with a positive exponent. Then, solve. Round your answer to four decimal places.
Question 1.
a. 3-2 = _____
Answer: 0.1111
When a power includes a negative exponent, express the number as 1 divided by the base, and change the exponent to positive.
3-2 = \(\frac{1}{3^{2}}\)
= \(\frac{1}{9}\)
= 0.1111
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. 8-2 = _____
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.
8-2 = \(\frac{1}{8^{2}}\)
= \(\frac{1}{64}\)
= 0.015625
Question 2.
a. 7-3 = _____
Answer: 0.002915
When a power includes a negative exponent, express the number as 1 divided by the base, and change the exponent to positive.
7-3 = \(\frac{1}{7^{3}}\)
= \(\frac{1}{343}\)
= 0.002915
b. 3-3 = _____
Answer: 0.037
When a power includes a negative exponent, express the number as 1 divided by the base, and change the exponent to positive.
3-3 = \(\frac{1}{3^{3}}\)
= \(\frac{1}{27}\)
= 0.037
c. 9-2 = _____
Answer: 0.01234
When a power includes a negative exponent, express the number as 1 divided by the base, and change the exponent to positive.
9-2 = \(\frac{1}{9^{2}}\)
= \(\frac{1}{81}\)
= 0.01234
Question 3.
a. 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
b. 5-2 = _____
Answer: 0.04
When a power includes a negative exponent, express the number as 1 divided by the base, and change the exponent to positive.
5-2 = \(\frac{1}{5^{2}}\)
= \(\frac{1}{25}\)
= 0.04
c. 2-3 = _____
Answer: 0.125
When a power includes a negative exponent, express the number as 1 divided by the base, and change the exponent to positive.
2-3 = \(\frac{1}{2^{3}}\)
= \(\frac{1}{8}\)
= 0.125
Question 4.
a. 2-4 = _____
Answer: 0.0625
When a power includes a negative exponent, express the number as 1 divided by the base, and change the exponent to positive.
2-4 = \(\frac{1}{2^{4}}\)
= \(\frac{1}{16}\)
= 0.0625
b. 10-3 = _____
Answer: 0.001
When a power includes a negative exponent, express the number as 1 divided by the base, and change the exponent to positive.
10-3= \(\frac{1}{10^{3}}\)
= \(\frac{1}{1000}\)
= 0.001
c. 1-4 = _____
Answer: 1
When a power includes a negative exponent, express the number as 1 divided by the base, and change the exponent to positive.
1-4= \(\frac{1}{1^{4}}\)
= \(\frac{1}{1}\)
= 1
Find each product. Round your answer to five decimal places.
Question 5.
a. 4-2× 4-3 = ____
Answer: 0.000976
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.
4-2× 4-3 = 4-5
By simplification,
= \(\frac{1}{4^{5}}\)
= \(\frac{1}{1024}\)
= 0.000976
b. 2-4× 2-1 = ____
Answer: 0.03125
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.
2-4× 2-1 = 2-5
By simplification,
= \(\frac{1}{2^{5}}\)
= \(\frac{1}{32}\)
= 0.03125
c. 3-2× 3-3 = ____
Answer: 0.004115
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.
3-2× 3-3 = 3-5
By simplification,
= \(\frac{1}{3^{5}}\)
= \(\frac{1}{243}\)
= 0.004115
Question 6.
a. 6-2× 6-2 = ____
Answer: 0.0007716
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.
6-2× 6-2 = 6-4
By simplification,
= \(\frac{1}{6^{4}}\)
= \(\frac{1}{1296}\)
= 0.0007716
b. 5-2× 5-4 = ____
Answer: 0.000064
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.
5-2× 5-4 = 5-6
By simplification,
= \(\frac{1}{5^{6}}\)
= \(\frac{1}{15625}\)
= 0.000064
c. 3-2× 3-2 = ____
Answer: 0.01234
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.
3-2× 3-2 = 3-4
By simplification,
= \(\frac{1}{3^{4}}\)
= \(\frac{1}{81}\)
= 0.01234
Question 7.
a. 8-6× 84 = ____
Answer: 0.015625
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.
8-6× 84 = 8-2
By simplification,
= \(\frac{1}{8^{2}}\)
= \(\frac{1}{64}\)
= 0.015625
b. 7-5× 72 = ____
Answer: 0.002915
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.
7-5× 72 = 7-3
By simplification,
= \(\frac{1}{7^{3}}\)
= \(\frac{1}{343}\)
= 0.002915
c. 2-7× 24 = ____
Answer: 0.125
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.
2-7× 24 = 2-3
By simplification,
= \(\frac{1}{2^{3}}\)
= \(\frac{1}{8}\)
= 0.125
Find each quotient. Round your answer to five decimal places.
Question 8.
a. 4-4 ÷ 4-2 = ____
Answer: 0.0625
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.
4-4 ÷ 4-2 = 4-2
By simplification,
= \(\frac{1}{4^{2}}\)
= \(\frac{1}{16}\)
= 0.0625
b. 8-5 ÷ 8-3 = ____
Answer: 0.015625
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.
8-5 ÷ 8-3 = 8-2
By simplification,
= \(\frac{1}{8^{2}}\)
= \(\frac{1}{64}\)
= 0.015625
c. 3-5 ÷ 3-2 = ____
Answer: 0.037
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.
3-5 ÷ 3-2 = 3-3
By simplification,
= \(\frac{1}{3^{3}}\)
= \(\frac{1}{27}\)
= 0.037
Question 9.
a. 2-8 ÷ 2-4 = ____
Answer: 0.0625
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.
2-8 ÷ 2-4 = 2-4
By simplification,
= \(\frac{1}{2^{4}}\)
= \(\frac{1}{16}\)
= 0.0625
b. 5-6 ÷ 5-4 = ____
Answer: 0.04
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.
5-6 ÷ 5-4 = 5-2
By simplification,
= \(\frac{1}{5^{2}}\)
= \(\frac{1}{25}\)
= 0.04
c. 6-7 ÷ 6-4 = ____
Answer: 0.004629
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.
6-7 ÷ 6-4 = 6-3
By simplification,
= \(\frac{1}{6^{3}}\)
= \(\frac{1}{216}\)
= 0.004629
Question 10.
a. 3-3 ÷ 32 = ____
Answer: 0.333
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.
3-3 ÷ 32 = 3-1
By simplification,
= \(\frac{1}{3^{1}}\)
= \(\frac{1}{3}\)
= 0.333
b. 4-3 ÷ 41 = ____
Answer: 0.0625
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.
4-3 ÷ 41 = 4-2
By simplification,
= \(\frac{1}{4^{2}}\)
= \(\frac{1}{16}\)
= 0.0625
c. 2-6 ÷ 2-3 = ____
Answer: 0.125
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.
2-6 ÷ 2-3 = 2-3
By simplification,
= \(\frac{1}{2^{3}}\)
= \(\frac{1}{8}\)
= 0.125
Rewrite each multiplication or division expression using a base and an exponent.
Question 1.
a. 3-4 × 3-6 = ____
Answer: 0.0000169
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.
3-4 × 3-6 = 3-10
By simplification,
= \(\frac{1}{3^{10}}\)
= \(\frac{1}{59049}\)
= 0.0000169
b. 9-3 ÷ 9-5 = ____
Answer: 81
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.
9-3 ÷ 9-5 = 92
By simplification,
= 81
Question 2.
a. 43 ÷ 4-2 = ____
Answer: 1024
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.
43 ÷ 4-2 = 45
By simplification,
= 1024
b. 55 × 5-6 = ____
Answer: 0.2
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.
55 × 5-6 = 5-1
By simplification,
= \(\frac{1}{5^{1}}\)
= \(\frac{1}{5}\)
= 0.2
Question 3.
a. 12-3 × 12-4 = ____
Answer: 0.00000002791
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.
12-3 × 12-4 = 12-7
By simplification,
= \(\frac{1}{12^{7}}\)
= \(\frac{1}{35831808}\)
= 0.00000002791
b. 4-6 × 44 = ____
Answer: 0.0625
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.
4-6 × 44 = 4-2
By simplification,
= \(\frac{1}{4^{2}}\)
= \(\frac{1}{16}\)
= 0.0625
Question 4.
a. 76 ÷ 7-3 = ____
Answer: 40353607
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.
76 ÷ 7-3 = 79
By simplification,
=40353607
b. 2-3 ÷ 23 = ____
Answer: 0.015625
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.
2-3 ÷ 23 = 2-6
By simplification,
=\(\frac{1}{2^{6}}\)
= \(\frac{1}{64}\)
= 0.015625
Question 5.
a. 114 × 11-3 = ____
Answer: 0.0909
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.
114 × 11-3 = 11-1
By simplification,
= \(\frac{1}{11^{1}}\)
= \(\frac{1}{11}\)
= 0.0909
b. 6-5 × 6-4 = ____
Answer: 0.00000009923
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.
6-5 × 6-4 = 6-9
By simplification,
= \(\frac{1}{6^{9}}\)
= \(\frac{1}{10077696}\)
= 0.00000009923
Question 6.
a. 8-5 ÷ 83 = ____
Answer: 0.0000000596
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.
8-5 ÷ 83 = 8-8
By simplification,
= \(\frac{1}{8^{8}}\)
= \(\frac{1}{16777216}\)
= 0.0000000596
b. 12-4 ÷ 12 = ____
Answer: 0.000004018
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.
12-4 ÷ 12 = 12-5
By simplification,
= \(\frac{1}{12^{5}}\)
= \(\frac{1}{248832}\)
= 0.000004018
Question 7.
a. 75 × 7-4 = ____
Answer: 7
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.
75 × 7-4 = 71
By simplification,
= 7
b. 5-3 × 52 = ____
Answer: 0.2
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.
5-3 × 52 = 5-1
By simplification,
= \(\frac{1}{5^{1}}\)
= \(\frac{1}{5}\)
= 0.2
Question 8.
a. 25 ÷ 2-3 = ____
Answer: 256
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.
25 ÷ 2-3 = 28
By simplification,
= 256
b. 3-12 × 3-4 = ____
Answer:
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.
6-5 × 6-4 = 6-9
By simplification,
= \(\frac{1}{6^{9}}\)
= \(\frac{1}{10077696}\)
= 0.00000009923
Question 9.
a. 63 ÷ 6-4 = ____
Answer: 279936
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.
63 ÷ 6-4 = 67
By simplification,
= 279936
b. 7-3 ÷ 74 = ____
Answer: 0.000001214
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.
7-3 ÷ 74 = 7-7
By simplification,
= \(\frac{1}{7^{7}}\)
= \(\frac{1}{823543}\)
= 0.000001214
Question 10.
a. 9-3 × 94 = ____
Answer: 9
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.
9-3 × 94 = 91
By simplification,
= 9
b. 10-5 × 10-2 = ____
Answer: 0.0000001
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.
10-5 × 10-2 = 10-7
By simplification,
= \(\frac{1}{10^{7}}\)
= \(\frac{1}{10000000}\)
= 0.0000001
Question 11.
a. 8-4 ÷ 8-2 = ____
Answer: 0.015625
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.
8-4 ÷ 8-2 = 8-2
By simplification,
= \(\frac{1}{8^{2}}\)
= \(\frac{1}{64}\)
= 0.015625
b. 2-2 × 2-12 = ____
Answer: 0.000061035
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.
2-2 × 2-12 = 2-14
By simplification,
= \(\frac{1}{2^{14}}\)
= \(\frac{1}{16384}\)
= 0.000061035
Question 12.
a. 3-6 × 3-3 = ____
Answer: 0.0000508
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.
3-6 × 3-3 = 3-9
By simplification,
= \(\frac{1}{3^{9}}\)
= \(\frac{1}{19683}\)
= 0.0000508
b. 8-6 ÷ 84 = ____
Answer: 0.0000000009313
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 ÷ 84 = 8-10
By simplification,
= \(\frac{1}{8^{10}}\)
= \(\frac{1}{1073741824}\)
= 0.0000000009313
Question 13.
a. 10-2 ÷ 103 = ____
Answer: 0.00001
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.
10-2 ÷ 103 = 10-5
By simplification,
= \(\frac{1}{10^{5}}\)
= \(\frac{1}{100000}\)
= 0.00001
b. 4-5 × 4-2 = ____
Answer: 0.000061
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.
4-5 × 4-2 = 4-7
By simplification,
= \(\frac{1}{4^{7}}\)
= \(\frac{1}{16384}\)
= 0.000061
Question 14.
a. 9-6 ÷ 9-3 = ____
Answer: 0.00137
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.
9-6 ÷ 9-3 = 9-3
By simplification,
= \(\frac{1}{9^{3}}\)
= \(\frac{1}{729}\)
= 0.00137
b. 114 ÷ 11-2 = ____
Answer: 1771561
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.
114 ÷ 11-2 = 116
By simplification,
= 1771561
Question 15.
a. 6-5 ÷ 63 = ____
Answer: 0.000000596
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.
6-5 ÷ 63 = 6-8
By simplification,
= \(\frac{1}{6^{8}}\)
= \(\frac{1}{1679616}\)
= 0.000000596
b. 5-12 × 5-4 = ____
Answer: 0.0000000000065536
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.
5-12 × 5-4 = 5-16
By simplification,
= \(\frac{1}{5^{16}}\)
= \(\frac{1}{152587890625}\)
= 0.0000000000065536
Question 16.
a. 12-6 ÷ 12 = ____
Answer: 0.00000002791
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.
12-6 ÷ 12 = 12-7
By simplification,
= \(\frac{1}{12^{7}}\)
= \(\frac{1}{35831808}\)
= 0.00000002791
b. 4-4 × 4-3 = ____
Answer: 0.000061
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.
4-4 × 4-3 = 4-7
By simplification,
= \(\frac{1}{4^{7}}\)
= \(\frac{1}{16384}\)
= 0.000061