Students can use the Spectrum Math Grade 8 Answer Key Chapter 1 Lesson 1.1 Using Exponents as a quick guide to resolve any of their doubts.
Spectrum Math Grade 8 Chapter 1 Lesson 1.1 Using Exponents Answers Key
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.
55 means 5 is used as a factor 5 times.
5 × 5 × 5 × 5 × 5 = 3,125 55 = 3,125
Write each power as a product of the factors.
Question 1.
a. 33 _____
Answer: 27
A power of a number represents repeated multiplication of the number by itself.
33 = 3 x 3 x 3 and is read 3 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 33, 3 is the base and 3 is the exponent.
33 means 3 is used as a factor 3 times.
3 x 3 x 3 = 27
33 = 27
b. 55 ______
Answer: 3125
A power of a number represents repeated multiplication of the number by itself.
55 = 5 × 5 × 5 × 5 x 5 and is read 5 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 55, 5 is the base and 5 is the exponent.
55 means 5 is used as a factor 5 times.
5 × 5 × 5 × 5 × 5 = 3,125
55 = 3,125
c. 61 _____
Answer: 6
A power of a number represents repeated multiplication of the number by itself.
61 = 6 and is read 6 to the first 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 61, 6 is the base and 1 is the exponent.
61 means 6 is used as a factor 1 times.
6 = 6
61 = 6
Question 2.
a. 212 = _____
Answer: 4096
A power of a number represents repeated multiplication of the number by itself.
212 = 2 × 2 × 2 × 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 and is read 2 to the twelfth 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 212, 2 is the base and 12 is the exponent.
212 means 2 is used as a factor 12 times.
2 × 2 × 2 × 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 = 3,125
212 = 3,125
b. 38 = _____
Answer: 6561
A power of a number represents repeated multiplication of the number by itself.
38 = 3 x 3 x 3 x 3 x 3 x 3 x 3 x 3 and is read 3 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 38, 3 is the base and 8 is the exponent.
38 means 3 is used as a factor 8 times.
3 x 3 x 3 x 3 x 3 x 3 x 3 x 3 = 6561
38 = 6561
c. 36 = _____
Answer: 729
A power of a number represents repeated multiplication of the number by itself.
36 = 3 x 3 x 3 x 3 x 3 x 3 and is read 3 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 36, 3 is the base and 6 is the exponent.
36 means 3 is used as a factor 6 times.
3 x 3 x 3 x 3 x 3 x 3 = 729
36 = 729
Question 3.
a. 47 = _____
Answer: 16384
A power of a number represents repeated multiplication of the number by itself.
47 = 4 x 4 x 4 x 4 x 4 x 4 x 4 and is read 4 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 47, 4 is the base and 7 is the exponent.
47 means 4 is used as a factor 7 times.
4 x 4 x 4 x 4 x 4 x 4 x 4 = 16384
47 = 16384
b. 44 = _____
Answer: 256
A power of a number represents repeated multiplication of the number by itself.
44 = 4 x 4 x 4 x 4 and is read 4 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 44 , 4 is the base and 4 is the exponent.
44 means 4 is used as a factor 4 times.
4 x 4 x 4 x 4 = 256
44 = 256
c. 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 = 512
83 = 512
Use exponents to rewrite these expressions.
Question 4.
a. 24 × 24 × 24 ____
Answer: 24 x 24 x 24 = 243
A power of a number represents repeated multiplication of the number by itself.
243 = 24 x 24 x 24 and is read 24 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 243, 24 is the base and 3 is the exponent.
243 means 24 is used as a factor 3 times.
24 x 24 x 24 = 243
24 x 24 x 24 = 7,077,888
243 = 7,077,888
b. 2 × 2 × 2 × 2 ______
Answer: 2 × 2 × 2 × 2 = 24
A power of a number represents repeated multiplication of the number by itself.
24 = 2 x 2 x 2 x 2 and is read 2 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 24, 2 is the base and 4 is the exponent.
24 means 2 is used as a factor 4 times.
2 × 2 × 2 × 2 = 24
2 × 2 × 2 × 2 = 16
24 = 16
Question 5.
a. 3 × 3 × 3 × 3 × 3 ______
Answer: 3 × 3 × 3 × 3 × 3 = 35
A power of a number represents repeated multiplication of the number by itself.
35 = 3 × 3 × 3 × 3 × 3 and is read 3 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 35, 3 is the base and 5 is the exponent.
35 means 3 is used as a factor 5 times.
3 × 3 × 3 × 3 × 3 = 35
3 × 3 × 3 × 3 × 3 = 243
35 = 243
b. 5 × 5 _____
Answer: 5 × 5 = 5²
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 6.
a. 5 × 5 × 5 × 5 × 5 × 5 ________
Answer: 5 × 5 × 5 × 5 × 5 × 5 = 56
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
b. 4 × 4 × 4 ____
Answer: 4 × 4 × 4 = 43
A power of a number represents repeated multiplication of the number by itself.
43 = 4 × 4 × 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 × 4 × 4 = 43
4 × 4 × 4 = 64
43 = 64
Find the value of each expression.
Question 7.
a. 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
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. 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
Write each power as a product of the factors.
Question 1.
a. 92 ______
Answer: 9 x 9
A power of a number represents repeated multiplication of the number by itself.
9² =9 x 9 and is read 9 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 9², 9 is the base and 2 is the exponent.
9² means 9 is used as a factor 2 times.
9 x 9 = 9²
9 x 9 = 81
9² = 81
b. 581 ______
Answer: 58
A power of a number represents repeated multiplication of the number by itself.
581 = 58 and is read 58 to the first 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 581, 58 is the base and 1 is the exponent.
581 means 58 is used as a factor 1 times.
58 = 581
58 = 58
581 = 58
c. 43 ______
Answer: 4 x 4 x 4
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. 54 ______
Answer: 5 x 5 x 5 x 5
A power of a number represents repeated multiplication of the number by itself.
54 = 5 x 5 x 5 x 5 and is read 5 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 54, 5 is the base and 4 is the exponent.
54 means 5 is used as a factor 4 times.
5 x 5 x 5 x 5 = 54
5 x 5 x 5 x 5 = 625
54 = 625
b. 82 ______
Answer: 8 x 8
A power of a number represents repeated multiplication of the number by itself.
82 = 8 x 8 and is read 8 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 82, 8 is the base and 2 is the exponent.
82 means 8 is used as a factor 2 times.
8 x 8 = 82
8 x 8 = 64
82 = 64
c. 34 ______
Answer: 3 x 3 x 3 x 3
A power of a number represents repeated multiplication of the number by itself.
34 = 3 x 3 x 3 x 3 and is read 3 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 34, 3 is the base and 4 is the exponent.
34 means 3 is used as a factor 4 times.
3 x 3 x 3 x 3 = 34
3 x 3 x 3 x 3 = 81
34 = 81
Question 3.
a. 752 ______
Answer: 75 x 75
A power of a number represents repeated multiplication of the number by itself.
752 = 75 x 75 and is read 75 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 752, 75 is the base and 2 is the exponent.
752 means 75 is used as a factor 2 times.
75 x 75 = 752
75 x 75 = 5625
752 = 5625
b. 62 ______
Answer: 6 x 6
A power of a number represents repeated multiplication of the number by itself.
62 = 6 x 6 and is read 6 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 62, 6 is the base and 2 is the exponent.
62 means 6 is used as a factor 2 times.
6 x 6 = 62
6 x 6 = 36
62 = 36
c. 1010 ______
Answer: 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10
A power of a number represents repeated multiplication of the number by itself.
1010 = 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 and is read 10 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 1010, 10 is the base and 10 is the exponent.
1010 means 10 is used as a factor 10 times.
10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 = 1010
10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 = 10,000,000,000
1010 = 10,000,000,000
Use exponents to rewrite these expressions.
Question 4.
a. 8 ______
Answer: 81
A power of a number represents repeated multiplication of the number by itself.
81 = 8 and is read 8 to the first 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 81, 8 is the base and 1 is the exponent.
81 means 8 is used as a factor 1 times.
8 = 81
8 = 8
81 = 8
b. 13 × 13 _____
Answer: 132
A power of a number represents repeated multiplication of the number by itself.
132 = 13 x 13 and is read 13 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 132, 13 is the base and 2 is the exponent.
132 means 13 is used as a factor 2 times.
13 x 13 = 132
13 x 13 = 169
132 = 169
Question 5.
a. 6 × 6 × 6 × 6 _____
Answer: 64
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
b. 5 × 5 × 5 × 5 _____
Answer: 54
A power of a number represents repeated multiplication of the number by itself.
54 = 5 × 5 × 5 × 5 and is read 5 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 54, 5 is the base and 4 is the exponent.
54 means 5 is used as a factor 4 times.
5 × 5 × 5 × 5 = 54
5 × 5 × 5 × 5 = 625
54 = 625
Question 6.
a. 2 × 2 × 2 × 2 × 2 × 2 × 2 _____
Answer: 27
A power of a number represents repeated multiplication of the number by itself.
27 =2 × 2 × 2 × 2 × 2 × 2 × 2 and is read 2 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 27, 2 is the base and 7 is the exponent.
27 means 2 is used as a factor 7 times.
2 × 2 × 2 × 2 × 2 × 2 × 2 = 27
2 × 2 × 2 × 2 × 2 × 2 × 2 = 128
27 = 128
b. 3 × 3 × 3 ____
Answer: 33
A power of a number represents repeated multiplication of the number by itself.
33 = 3 × 3 × 3 and is read 3 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 33, 3 is the base and 3 is the exponent.
33 means 3 is used as a factor 3 times.
3 × 3 × 3 = 33
3 × 3 × 3 = 27
33 = 27
Question 7.
a. 86 × 86 × 86 _____
Answer: 863
A power of a number represents repeated multiplication of the number by itself.
863 = 86 × 86 × 86 and is read 86 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 863, 86 is the base and 3 is the exponent.
33 means 86 is used as a factor 3 times.
86 × 86 × 86 = 863
86 × 86 × 86 = 636,056
863 = 636,056
b. 4 × 4 × 4 × 4 × 4 _____
Answer: 45
A power of a number represents repeated multiplication of the number by itself.
45 = 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 45, 4 is the base and 5 is the exponent.
45 means 4 is used as a factor 5 times.
4 × 4 × 4 × 4 × 4 = 45
4 × 4 × 4 × 4 × 4 = 1024
45 = 1024
Question 8.
a. 10 × 10 × 10 × 10 × 10 _____
Answer: 105
A power of a number represents repeated multiplication of the number by itself.
105 = 10 × 10 × 10 × 10 × 10 and is read 10 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 105 , 10 is the base and 5 is the exponent.
105 means 10 is used as a factor 5 times.
10 × 10 × 10 × 10 × 10 = 105
10 × 10 × 10 × 10 × 10 = 100000
105 = 100000
b. 15 × 15 × 15 × 15 × 15 ____
Answer:155
A power of a number represents repeated multiplication of the number by itself.
155 = 15 × 15 × 15 × 15 × 15 and is read 15 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 155 , 15 is the base and 5 is the exponent.
155 means 15 is used as a factor 5 times.
15 × 15 × 15 × 15 × 15 = 155
15 × 15 × 15 × 15 × 15 = 759,375
155 = 759,375
Find the value of each expression.
Question 9.
a. 71 = _____
Answer: 7
A power of a number represents repeated multiplication of the number by itself.
71 = 7 and is read 7 to the first 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 71, 7 is the base and 1 is the exponent.
71 means 7 is used as a factor 1 times.
7 = 71
71 = 7
b. 34 = ____
Answer: 81
A power of a number represents repeated multiplication of the number by itself.
34 = 3 x 3 x 3 x 3 and is read 3 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 34, 3 is the base and 4 is the exponent.
34 means 3 is used as a factor 4 times.
3 x 3 x 3 x 3 = 34
3 x 3 x 3 x 3 = 81
34 = 81
c. 105 = ____
Answer: 100000
A power of a number represents repeated multiplication of the number by itself.
105 = 10 × 10 × 10 × 10 × 10 and is read 10 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 105 , 10 is the base and 5 is the exponent.
105 means 10 is used as a factor 5 times.
10 × 10 × 10 × 10 × 10 = 105
10 × 10 × 10 × 10 × 10 = 100000
105 = 100000
Question 10.
a. 75 = ____
Answer: 16,807
A power of a number represents repeated multiplication of the number by itself.
75 = 7 x 7 x 7 x 7 x 7 and is read 7 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 75 , 7 is the base and 5 is the exponent.
75 means 7 is used as a factor 5 times.
7 x 7 x 7 x 7 x 7 = 75
7 x 7 x 7 x 7 x 7 = 16,807
75 = 16,807
b. 53 = ____
Answer: 125
A power of a number represents repeated multiplication of the number by itself.
53 = 5 x 5 x 5 and is read 5 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 53, 5 is the base and 3 is the exponent.
53 means 5 is used as a factor 3 times.
5 x 5 x 5 = 53
5 x 5 x 5 = 125
53 = 125
c. 84 = ____
Answer: 4096
A power of a number represents repeated multiplication of the number by itself.
84 = 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 84, 8 is the base and 4 is the exponent.
84 means 8 is used as a factor 4 times.
8 x 8 x 8 x 8 = 84
8 x 8 x 8 x 8 = 4096
84 = 4096
Question 11.
a. 42 = ____
Answer: 16
A power of a number represents repeated multiplication of the number by itself.
42 = 4 x 4 and is read 4 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 42, 4 is the base and 2 is the exponent.
42 means 4 is used as a factor 2 times.
4 x 4 = 42
4 x 4 = 16
42 = 16
b. 25 = ____
Answer: 32
A power of a number represents repeated multiplication of the number by itself.
25 = 2 x 2 x 2 x 2 x 2 and is read 2 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 25 , 2 is the base and 5 is the exponent.
25 means 2 is used as a factor 5 times.
2 x 2 x 2 x 2 x 2 = 25
2 x 2 x 2 x 2 x 2 = 32
c. 97 = ____
Answer: 4,782,969
A power of a number represents repeated multiplication of the number by itself.
97 = 9 x 9 x 9 x 9 x 9 x 9 x 9 and is read 9 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 97, 9 is the base and 7 is the exponent.
97 means 9 is used as a factor 7 times.
9 x 9 x 9 x 9 x 9 x 9 x 9 = 97
9 x 9 x 9 x 9 x 9 x 9 x 9 = 4,782,969
97 = 4,782,969
Question 12.
a. 64 = ____
Answer: 1296
A power of a number represents repeated multiplication of the number by itself.
64 = 6 x 6 x 6 x 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 x 6 x 6 x 6 = 64
6 x 6 x 6 x 6 = 1296
64 = 1296
b. 123 = ____
Answer: 1728
A power of a number represents repeated multiplication of the number by itself.
123 = 12 x 12 x 12 and is read 12 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 123, 12 is the base and 3 is the exponent.
123 means 12 is used as a factor 3 times.
12 x 12 x 12 = 123
12 x 12 x 12 = 1728
123 = 1728
c. 73 = ____
Answer: 343
A power of a number represents repeated multiplication of the number by itself.
73 = 7 x 7 x 7 and is read 7 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 73, 7 is the base and 3 is the exponent.
73 means 7 is used as a factor 3 times.
7 x 7 x 7 = 73
7 x 7 x 7 = 343
73 = 343