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. 37 = ____
Answer: 2187
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 = ____
Answer: 65536
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 = ____
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. 129 = ____
Answer: 5159780352
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 = ____
Answer: 1024
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 = ____
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 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. 74 = ____
Answer: 2401
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 = ____
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. 107 = ____
Answer: 10000000
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 = ____
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. 28 = ____
Answer: 256
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 = _____
Answer: 32768
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 = _____
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. 62 × 64 = _____
Answer: 46656
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 = _____
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 × 43 = 42
By simplification,
= 16
b. 34 ÷ 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.
34 ÷ 3-3 = 37
By simplification,
= 2187
c. 12-2 ÷ 124 = ____
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 ÷ 124 = 12-6
By simplification,
= \(\frac{1}{12^{6}}\)
= \(\frac{1}{2985984}\)
= 0.000000334
Question 9.
a. 54 × 57 = ____
Answer: 48828125
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 = ____
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. 58 × 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.
58 × 5-3 = 55
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. 78 ÷ 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.
78 ÷ 7-3= 711
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 = 62
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. 94 ÷ 98 = ____
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.
94 ÷ 98 = 912
By simplification,
= 282429536481
c. 3-8 ÷ 34 = ____
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 ÷ 34 = 3-12
By simplification,
= \(\frac{1}{3^{10}}\)
= \(\frac{1}{531441}\)
= 0.00000188
Question 12.
a. 10-3 × 103 = ____
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 × 103 = 100
By simplification,
= 1
b. 86 ÷ 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.
86 ÷ 8-3 = 89
By simplification,
= 134217728
c. 74 × 72 = ____
Answer: 117649
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
__________________
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 × 102
__________________
Answer: 426
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
__________________
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 × 104
__________________
Answer: 65000
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
__________________
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 × 106
__________________
Answer: 8273400
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
__________________
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 × 107
__________________
Answer: 82300000
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
__________________
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 × 105
__________________
Answer:238200
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
__________________
Answer: 91200000
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
__________________
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× 103
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
__________________
Answer: 1.032× 101
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
__________________
Answer: 4.141× 102
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
__________________
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× 101
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
__________________
Answer: 8.524× 103
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
__________________
Answer: 2.39390× 105
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
__________________
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× 104
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
__________________
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: 3.476000×106
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
__________________
Answer: 3.7120000× 107
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
__________________
Answer: 3.74200× 105
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× 105
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