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

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Integers and Exponents

Find the value of each expression.

Question 1.
a. 37 = ____
A power of a number represents repeated multiplication of the number by itself.
37 = 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 37, 3 is the base and 7 is the exponent.
37 means 3 is used as a factor 7 times.
3 x 3 x 3 x 3 x 3 x 3 x 3 = 37
3 x 3 x 3 x 3 x 3 x 3 x 3 = 2187
37 = 2187

b. 48 = ____
A power of a number represents repeated multiplication of the number by itself.
48 = 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 48, 4 is the base and 8 is the exponent.
48 means 4 is used as a factor 8 times.
4 x 4  x 4 x 4 x 4 x 4 x 4 x 4 = 48
4 x 4  x 4 x 4 x 4 x 4 x 4 x 4 = 65536
48 = 65536

c. 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 2.
a. 129 = ____
A power of a number represents repeated multiplication of the number by itself.
129 = 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 129, 12 is the base and 9 is the exponent.
129 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 = 129
12 x 12 x 12 x 12 x 12 x 12 x 12 x 12 x 12 = 5159780352
125 = 5159780352

b. 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

c. 84 = ____
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 3.
a. 3-6 = ____
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 = ____
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 = ____
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 = ____
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 = ____
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 = ____
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 = ____
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. 74 = ____
A power of a number represents repeated multiplication of the number by itself.
74 = 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 74, 7 is the base and 4 is the exponent.
74 means 7 is used as a factor 4 times.
7 x 7 x 7 x 7 = 74
7 x 7 x 7 x 7 = 2401
74 = 2401

c. 3-9 = ____
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. 107 = ____
A power of a number represents repeated multiplication of the number by itself.
107 = 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 107, 10 is the base and 7 is the exponent.
107 means 10 is used as a factor 7 times.
10 x 10 x 10 x 10 x 10 x 10 x 10 = 107
10 x 10 x 10 x 10 x 10 x 10 x 10 = 10000000
37 = 10000000

b. 9-2 = ____
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. 28 = ____
A power of a number represents repeated multiplication of the number by itself.
28 = 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 28, 2 is the base and 8 is the exponent.
28 means 2 is used as a factor 8 times.
2 x 2 x 2 x 2  x 2 x 2 x 2 x 2 = 28
2 x 2 x 2 x 2  x 2 x 2 x 2 x 2 = 256
28 = 256

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

Question 7.
a. 82 × 83 = _____
To multiply powers with the same base, combine bases, add the exponents, then simplify.
82 × 83 = 82 + 3 = 85 = 32768

b. 5-5 × 5-2 = _____
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.
5-5 × 5-2 = 5-7
By simplification,
= $$\frac{1}{5^{7}}$$
= $$\frac{1}{78125}$$
= 0.0000128

c. 62 × 64 = _____
To multiply powers with the same base, combine bases, add the exponents, then simplify.
62 × 64 = 62 + 4 = 66 = 46656

Question 8.
a. 4-1 × 43 = _____
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 × 43 = 42
By simplification,
= 16

b. 34 ÷ 3-3 = ____
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.
34 ÷ 3-3 = 37
By simplification,
= 2187

c. 12-2 ÷ 124 = ____
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 ÷ 124 = 12-6
By simplification,
= $$\frac{1}{12^{6}}$$
= $$\frac{1}{2985984}$$
= 0.000000334

Question 9.
a. 54 × 57 = ____
To multiply powers with the same base, combine bases, add the exponents, then simplify.
54 × 57 = 54 + 7 = 511 = 48828125

b. 8-2 × 8-6 = ____
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. 58 × 5-3 = ____
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.
58 × 5-3 = 55
By simplification,
= $$\frac{1}{5^{7}}$$
= $$\frac{1}{3125}$$
= 0.00032

Question 10.
a. 9-2 × 9-5 = ____
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. 78 ÷ 7-3 = ____
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.
78 ÷ 7-3= 711
By simplification,
= 1977326743

c. 6-2 ÷ 6-4 = ____
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 = 62
By simplification,
= 36

Question 11.
a. 7-1 × 7-3 = ____
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. 94 ÷ 98 = ____
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.
94 ÷ 98 = 912
By simplification,
= 282429536481

c. 3-8 ÷ 34 = ____
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 ÷ 34 = 3-12
By simplification,
= $$\frac{1}{3^{10}}$$
= $$\frac{1}{531441}$$
= 0.00000188

Question 12.
a. 10-3 × 103 = ____
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 × 103 = 100
By simplification,
= 1

b. 86 ÷ 8-3 = ____
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.
86 ÷ 8-3 = 89
By simplification,
= 134217728

c. 74 × 72 = ____
To multiply powers with the same base, combine bases, add the exponents, then simplify.
74 × 72 = 74 + 2 = 76 = 117649

Rewrite each number in standard notation.

Question 13.
a. 3.04 × 10-3
__________________
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 × 102
__________________
Translate numbers written in scientific notation into standard form by reading the exponent.
Move the decimal right 2 times
4.26 × 102= 426

c. 8.1 × 10-4
__________________
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 × 104
__________________
Translate numbers written in scientific notation into standard form by reading the exponent.
Move the decimal right 4 times
6.5 × 104= 65000

b. 2.4 × 10-2
__________________
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
__________________
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
__________________
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 × 106
__________________
Translate numbers written in scientific notation into standard form by reading the exponent.
Move the decimal right 6 times
8.2734 × 106= 8273400

c. 7.362 × 10-6
__________________
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 × 107
__________________
Translate numbers written in scientific notation into standard form by reading the exponent.
Move the decimal right 7 times
8.23 × 107= 82300000

b. 4.602 × 10-3
__________________
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 × 105
__________________
Translate numbers written in scientific notation into standard form by reading the exponent.
Move the decimal right 5 times
2.382 × 105= 238200

Question 17.
a. 9.12 × 107
__________________
Translate numbers written in scientific notation into standard form by reading the exponent.
Move the decimal right 7 times
9.12 × 107= 91200000

b. 7.292 × 10-3
__________________
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
__________________
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
__________________
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× 103

b. 10.32
__________________
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× 101

c. 414.1
__________________
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 2 places and multiply the shift by the power of 10.
414.1 =4.141× 102

Question 19.
a. 0.0003954
__________________
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
__________________
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× 101

c. 8,524
__________________
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× 103

Question 20.
a. 239,390
__________________
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× 105

b. 0.003121
__________________
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
__________________
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× 104

Question 21.
a. 0.0283
__________________
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
__________________
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
__________________
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 = 3.476000×106

Question 22.
a. 37, 120,000
__________________
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× 107

b. 374,200
__________________