Students can use the Spectrum Math Grade 8 Answer Key Chapter 1 Lesson 1.2 Equivalent Expressions with Exponents as a quick guide to resolve any of their doubts.
Spectrum Math Grade 8 Chapter 1 Lesson 1.2 Equivalent Expressions with Exponents Answers Key
To multiply powers with the same base, combine bases, add the exponents, then simplify.
22 × 23 = 22 + 3 = 25 = 32
To divide powers with the same base, combine bases, subtract the exponents, then simplify.
35 ÷ 32 = 35 – 2 = 33 = 27
Find the value of each expression.
Question 1.
a. 72 = ____
Answer: 49
A power of a number represents repeated multiplication of the number by itself.
72 =7 x 7 and is read 7 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 72, 7 is the base and 2 is the exponent.
72 means 7 is used as a factor 2 times.
7 x 7 = 72
7 x 7 = 49
72 = 49
b. 83 = ____
Answer: 512
A power of a number represents repeated multiplication of the number by itself.
83 = 8 x 8 x 8 and is read 8 to the third 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 83 , 8 is the base and 3 is the exponent.
83 means 8 is used as a factor 3 times.
8 x 8 x 8 = 83
8 x 8 x 8 = 512
83 = 512
c. 43 = _____
Answer: 64
A power of a number represents repeated multiplication of the number by itself.
43 = 4 x 4 x 4 and is read 4 to the third 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 43, 4 is the base and 3 is the exponent.
43 means 4 is used as a factor 3 times.
4 x 4 x 4 = 43
4 x 4 x 4 = 64
43 = 64
Question 2.
a. 102 = ____
Answer: 100
A power of a number represents repeated multiplication of the number by itself.
102 =10 x 10 and is read 10 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 102 , 10 is the base and 2 is the exponent.
102 means 10 is used as a factor 2 times.
10 x 10 = 102
10 x 10 = 100
102 = 100
b. 94 = ____
Answer: 6561
A power of a number represents repeated multiplication of the number by itself.
94 = 9 x 9 x 9 x 9 and is read 9 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 94, 9 is the base and 4 is the exponent.
94 means 9 is used as a factor 4 times.
9 x 9 x 9 x 9 = 94
9 x 9 x 9 x 9 = 6561
94 = 6561
c. 115 = _____
Answer: 161051
A power of a number represents repeated multiplication of the number by itself.
115 = 11 x 11 x 11 x 11 x 11 and is read 11 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 115, 11 is the base and 5 is the exponent.
115 means 11 is used as a factor 5 times.
11 x 11 x 11 x 11 x 11 = 115
11 x 11 x 11 x 11 x 11 = 161051
115 = 161051
Question 3.
a. 173 = ____
Answer: 4913
A power of a number represents repeated multiplication of the number by itself.
173 = 17 x 17 x 17 and is read 17 to the third 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 173, 17 is the base and 3 is the exponent.
173 means 17 is used as a factor 3 times.
17 x 17 x 17 = 173
17 x 17 x 17 = 4913
173 = 4913
b. 56 = ____
Answer: 15625
A power of a number represents repeated multiplication of the number by itself.
56 = 5 × 5 × 5 × 5 × 5 × 5 and is read 5 to the sixth 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 56, 5 is the base and 6 is the exponent.
56 means 5 is used as a factor 6 times.
5 × 5 × 5 × 5 × 5 × 5 = 56
5 × 5 × 5 × 5 × 5 × 5 = 15625
56 = 15625
c. 64 = _____
Answer: 1296
A power of a number represents repeated multiplication of the number by itself.
64 = 6 × 6 × 6 × 6 and is read 6 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 64, 6 is the base and 4 is the exponent.
64 means 6 is used as a factor 4 times.
6 × 6 × 6 × 6 = 64
6 × 6 × 6 × 6 = 1296
64 = 1296
Question 4.
a. 213 = ____
Answer: 9261
A power of a number represents repeated multiplication of the number by itself.
213 = 21 x 21 x 21 and is read 21 to the third 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 213, 21 is the base and 3 is the exponent.
213 means 21 is used as a factor 3 times.
21 x 21 x 21 = 213
21 x 21 x 21 = 9261
213 = 9261
b. 164 = ____
Answer: 65536
A power of a number represents repeated multiplication of the number by itself.
164 = 16 × 16 × 16 × 16 and is read 16 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 164, 16 is the base and 4 is the exponent.
164 means 16 is used as a factor 4 times.
16 × 16 × 16 × 16 = 64
16 × 16 × 16 × 16 = 65536
64 = 65536
c. 125 = _____
Answer: 248832
A power of a number represents repeated multiplication of the number by itself.
125 = 12 x 12 x 12 x 12 x 12 and is read 12 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 125, 12 is the base and 5 is the exponent.
125 means 11 is used as a factor 5 times.
12 x 12 x 12 x 12 x 12 = 125
12 x 12 x 12 x 12 x 12 = 248832
125 = 248832
Rewrite each expression as one base and one exponent. Then, find the value.
Question 5.
a. 82 × 83 = ____
Answer: 32768
To multiply powers with the same base, combine bases, add the exponents, then simplify.
82 × 83 = 82 + 3 = 85 = 32768
b. 33 × 33 = ____
Answer: 729
To multiply powers with the same base, combine bases, add the exponents, then simplify.
33 × 33 = 33 + 3 = 36 = 729
c. 22 × 22 = ____
Answer: 16
To multiply powers with the same base, combine bases, add the exponents, then simplify.
22 × 22 = 22 + 2 = 24 = 16
Question 6.
a. 74 ÷ 72 = ___
Answer: 49
To divide powers with the same base, combine bases, subtract the exponents, then simplify.
74 ÷ 72 = 74 – 2 = 72 = 49
b. 95 ÷ 93 = ___
Answer: 81
To divide powers with the same base, combine bases, subtract the exponents, then simplify.
95 ÷ 93 = 95 – 3 = 92 = 81
c. 164 ÷ 162 = ___
Answer: 256
To divide powers with the same base, combine bases, subtract the exponents, then simplify.
164 ÷ 162 = 164 – 2 = 162 = 256
Question 7.
a. 64 × 61 = ____
Answer: 7776
To multiply powers with the same base, combine bases, add the exponents, then simplify.
64 × 61 = 64 + 1 = 65 = 7776
b. 44 × 42 = ____
Answer: 4096
To multiply powers with the same base, combine bases, add the exponents, then simplify.
44 × 42 = 44 + 2 = 46 = 4096
c. 32 × 32 = ____
Answer: 81
To multiply powers with the same base, combine bases, add the exponents, then simplify.
32 × 32 = 32 + 2 = 34 = 81
Question 8.
a. 106 ÷ 104 = ____
Answer: 100
To divide powers with the same base, combine bases, subtract the exponents, then simplify.
106 ÷ 104 = 106 – 4 = 102 = 100
b. 83 ÷ 82 = ____
Answer: 8
To divide powers with the same base, combine bases, subtract the exponents, then simplify.
83 ÷ 82 = 83 – 2 = 81 = 8
c. 76 ÷ 73 = ____
Answer: 343
To divide powers with the same base, combine bases, subtract the exponents, then simplify.
76 ÷ 73 = 76 – 3 = 73 = 343
Question 9.
a. 53 × 52 = ____
Answer: 3125
To multiply powers with the same base, combine bases, add the exponents, then simplify.
53 × 52 = 53 + 2 = 55 = 3125
b. 103 × 104 = ____
Answer: 10000000
To multiply powers with the same base, combine bases, add the exponents, then simplify.
103 × 104 = 103 + 4 = 107 = 10000000
c. 152 × 151 = ____
Answer: 3375
To multiply powers with the same base, combine bases, add the exponents, then simplify.
152 × 151= 152 + 1 = 153 = 3375
d. 66 ÷ 63 = ____
Answer: 216
To divide powers with the same base, combine bases, subtract the exponents, then simplify.
66 ÷ 63 = 66 – 3 = 63 = 216
Rewrite each multiplication or division expression using a base and an exponent.
Question 1.
a. 43 × 45 = _____
Answer: 65536
To multiply powers with the same base, combine bases, add the exponents, then simplify.
43 × 45= 43 + 5 = 48 = 65536
b. 92 × 93 = _____
Answer: 59049
To multiply powers with the same base, combine bases, add the exponents, then simplify.
92 × 93 = 92 + 3 = 95 = 59049
Question 2.
a. (3 × 3 × 3) × (3 × 3) = _______
Answer: 33 × 32
33 + 2 = 35 = 243
To multiply powers with the same base, combine bases, add the exponents, then simplify.
(3 × 3 × 3) × (3 × 3) = 33 × 32 = 33 + 2 = 35 = 243
b. 56 ÷ 53 = _______
Answer: 125
To divide powers with the same base, combine bases, subtract the exponents, then simplify.
56 ÷ 53 = 56 – 3 = 53 = 125
Question 3.
a. 85 ÷ 8 = _______
Answer: 4096
To divide powers with the same base, combine bases, subtract the exponents, then simplify.
85 ÷ 8 = 85 – 1 = 84 = 4096
b. (2 × 2 × 2 × 2) ÷ (2 × 2) = _______
Answer: 24 × 22
24 + 2 = 26 = 64
To multiply powers with the same base, combine bases, add the exponents, then simplify.
(2 × 2 × 2 × 2) ÷ (2 × 2) = 24 × 22 = 24 + 2 = 26 = 64
Question 4.
a. (5 × 5) × (5 × 5) = ______
Answer: 52 × 52
52 + 2 = 54 = 625
To multiply powers with the same base, combine bases, add the exponents, then simplify.
(5 × 5) × (5 × 5)= 52 × 52 = 52 + 2 = 54 = 625
b. 99 ÷ 95 = _______
Answer: 6561
To divide powers with the same base, combine bases, subtract the exponents, then simplify.
99 ÷ 95 = 99 – 5 = 94 = 6561
Question 5.
a. 103 × 10 = ______
Answer: 1000
To multiply powers with the same base, combine bases, add the exponents, then simplify.
103 × 10 = 103 + 1 = 104 = 1000
b. 65 ÷ 62 = _______
Answer: 216
To divide powers with the same base, combine bases, subtract the exponents, then simplify.
65 ÷ 62 = 65 – 2 = 63 = 216
Question 6.
a. 43 ÷ 42 = _______
Answer: 4
To divide powers with the same base, combine bases, subtract the exponents, then simplify.
43 ÷ 42 = 43 – 2 = 41 = 4
b. (7 × 7 × 7) ÷ 7 = _______
Answer: 73 ÷ 71
73 – 1 = 72 = 49
To divide powers with the same base, combine bases, subtract the exponents, then simplify.
(7 × 7 × 7) ÷ 7 =73 ÷ 71= 73 – 1 = 72 = 49
Question 7.
a. 115 × 112 = _______
Answer: 19487171
To multiply powers with the same base, combine bases, add the exponents, then simplify.
115 × 112 = 115 + 2 = 117 = 19487171
b. 6 × 65 = _______
Answer: 46656
To multiply powers with the same base, combine bases, add the exponents, then simplify.
6 × 65 = 61 + 5 = 66 = 46656
Question 8.
a. (8 × 8 × 8 × 8) ÷ (8 × 8) = _____
Answer: 84 ÷ 82
84- 2 = 82 = 64
To divide powers with the same base, combine bases, subtract the exponents, then simplify.
(8 × 8 × 8 × 8) ÷ (8 × 8) =84 ÷ 82 =84- 2 = 82 = 64
b. 53 × 52 = ____
Answer: 3125
To multiply powers with the same base, combine bases, add the exponents, then simplify.
53 × 52 = 53 + 2 = 55 = 3125
Question 9.
a. 129 × 122 = ____
Answer: 743008370688
To multiply powers with the same base, combine bases, add the exponents, then simplify.
129 × 122 = 129 + 2 = 1211 = 743008370688
b. 1110 ÷ 114 = ____
Answer: 1771561
To divide powers with the same base, combine bases, subtract the exponents, then simplify.
1110 ÷ 114 = 1110 – 4 = 116 = 1771561
Question 10.
a. 34 × 34 = ____
Answer: 6561
To multiply powers with the same base, combine bases, add the exponents, then simplify.
34 × 34 = 34 + 4 = 38 = 6561
b. (4 × 4 × 4 × 4) ÷ 4 = ____
Answer: 64
To divide powers with the same base, combine bases, subtract the exponents, then simplify.
(4 × 4 × 4 × 4) ÷ 4 = 44 ÷ 41 = 44 – 1 = 43 = 64
Question 11.
a. (5 × 5 × 5) ÷ 5 = _____
Answer: 25
To divide powers with the same base, combine bases, subtract the exponents, then simplify.
(5 × 5 × 5) ÷ 5 =53 ÷ 51 =53- 1 = 52 = 25
b. 68 × 64 = _____
Answer: 2176782336
To multiply powers with the same base, combine bases, add the exponents, then simplify.
68 × 64 = 68 + 4 = 612 = 2176782336
Question 12.
a. 412 ÷ 46 = ____
Answer: 4096
To divide powers with the same base, combine bases, subtract the exponents, then simplify.
412 ÷ 46 = 412- 6 = 46 = 4096
b. 33 × 39 = ____
Answer: 531441
To multiply powers with the same base, combine bases, add the exponents, then simplify.
33 × 39 = 33 + 9 = 312 = 531441
Question 13.
a. (6 × 6 × 6 × 6) ÷ (6 × 6 × 6) = _____
Answer: 6
To divide powers with the same base, combine bases, subtract the exponents, then simplify.
(6 × 6 × 6 × 6) ÷ (6 × 6 × 6) = 64 ÷ 63 = 64 – 3 = 61 = 6
b. 158 ÷ 153 = _____
Answer: 759375
To divide powers with the same base, combine bases, subtract the exponents, then simplify.
158 ÷ 153 = 158 – 3 = 155 = 759375
Question 14.
a. 99 × 96 = ____
Answer: 205891132094649
To multiply powers with the same base, combine bases, add the exponents, then simplify.
99 × 96 = 99 + 6 = 915 = 205891132094649
b. 78 × 72 = ____
Answer: 282475249
To multiply powers with the same base, combine bases, add the exponents, then simplify.
78 × 72 = 78 + 2 = 710 = 282475249
Question 15.
a. 27 ÷ 2 = ____
Answer: 64
To divide powers with the same base, combine bases, subtract the exponents, then simplify.
27 ÷ 2 = 27 – 1 = 26 = 64
b. 411 × 4 = ____
Answer: 16777216
To multiply powers with the same base, combine bases, add the exponents, then simplify.
411 × 4 = 411 + 1 = 412 = 16777216