Spectrum Math Grade 8 Chapter 1 Pretest Answer Key

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

Spectrum Math Grade 8 Chapter 1 Pretest Answers Key

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Integers and Exponents

Find the value of each expression.

Question 1.
a. 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

b. 8⁵ = _____
Answer: 32768
A power of a number represents repeated multiplication of the number by itself.
8⁵  = 8 x 8 x 8 x 8 x 8 and is read 8 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 8⁵, 8 is the base and 5 is the exponent.
8⁵ means 8 is used as a factor 5 times.
8 x 8 x 8 x 8 x 8 = 8⁵
8 x 8 x 8 x 8 x 8 = 32768
8⁵ = 32768

c. 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

Question 2.
a. 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

b. 1⁵ = _____
Answer: 1
A power of a number represents repeated multiplication of the number by itself.
1⁵  = 1 x 1 x 1 x 1 x 1 and is read 1 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 1⁵, 1 is the base and 5 is the exponent.
1⁵ means 1 is used as a factor 5 times.
1 x 1 x 1 x 1 x 1 = 1⁵
1 x 1 x 1 x 1 x 1 = 1
1⁵ = 1

c. 6⁸ = _____
Answer: 1679616
A power of a number represents repeated multiplication of the number by itself.
6⁸ = 6 x 6 x 6 x 6 x 6 x 6 x 6 x 6 and is read 6 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 6⁸, 6 is the base and 8 is the exponent.
6⁸ means 6 is used as a factor 8 times.
6 x 6 x 6 x 6 x 6 x 6 x 6 x 6 = 6⁸
6 x 6 x 6 x 6 x 6 x 6 x 6 x 6 = 1679616
6⁸ = 1679616

Question 3.
a. 4^-3 = _____
Answer: 0.015626
When a power includes a negative exponent, express the number as 1 divided by the base, and change the exponent to positive.
4^-3 = \(\frac{1}{4^{3}}\)
= \(\frac{1}{64}\)
= 0.015626

b. 3^-5 = _____
Answer: 0.004115
When a power includes a negative exponent, express the number as 1 divided by the base, and change the exponent to positive.
3^-5= \(\frac{1}{3^{5}}\)
= \(\frac{1}{243}\)
= 0.004115

c. 7^-4 = _____
Answer: 0.000416
When a power includes a negative exponent, express the number as 1 divided by the base, and change the exponent to positive.
7^-4= \(\frac{1}{7^{4}}\)
= \(\frac{1}{2401}\)
= 0.000416

Question 4.
a. 2^-5 = _____
Answer: 0.03125
When a power includes a negative exponent, express the number as 1 divided by the base, and change the exponent to positive.
2^-5 = \(\frac{1}{2^{5}}\)
= \(\frac{1}{32}\)
= 0.03125

b. 9^-3 = _____
Answer: 0.001371
When a power includes a negative exponent, express the number as 1 divided by the base, and change the exponent to positive.
9^-3= \(\frac{1}{9^{3}}\)
= \(\frac{1}{729}\)
= 0.001371

c. 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

Question 5.
a. 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

b. 3^-4 = _____
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.
3^-4 = \(\frac{1}{3^{4}}\)
= \(\frac{1}{81}\)
= 0.012345

c. 5⁹ = _____
Answer: 1953125
A power of a number represents repeated multiplication of the number by itself.
5⁹ = 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 and is read 5 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 5⁹, 5 is the base and 9 is the exponent.
5⁹ means 5 is used as a factor 9 times.
5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 = 5⁹
5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 = 1953125
12⁵ = 1953125

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

Question 6.
a. 4⁵ ÷ 4² = ____
Answer: 16384
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.
4⁵ ÷ 4²= 4⁷
By simplification,
= 16384

b. 6^-5 × 6³ = ____
Answer: 0.02778
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 × 6³ = 6^-2
By simplification,
= \(\frac{1}{6^{2}}\)
= \(\frac{1}{36}\)
= 0.02778

c. 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

Question 7.
a. 9¹¹ ÷ 9⁶ = ____
Answer: 16677181699666569
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.
9¹¹ ÷ 9⁶=9¹⁷
By simplification,
= 16677181699666569

b. 5^-3 × 5^-1 = ____
Answer: 0.0016
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 × 5^-1 = 5^-4
By simplification,
= \(\frac{1}{6^{2}}\)
= \(\frac{1}{625}\)
= 0.0016

c. 3^-6 ÷ 3⁴ = ____
Answer: 0.0000169
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^-6 ÷ 3⁴ = 3^-10
By simplification,
= \(\frac{1}{3^{10}}\)
= \(\frac{1}{59049}\)
= 0.0000169

Question 8.
a. 8² × 8³ = ____
Answer: 32768
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.
8² × 8³ = 8⁵
By simplification,
= 32768

b. 6⁴ × 6⁷ = ____
Answer: 362797056
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.
6⁴ × 6⁷ = 6¹¹
By simplification,
= 362797056

c. 4^-2 ÷ 4^-5 = ____
Answer: 64
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^-2 ÷ 4^-5 = 4³
By simplification,
=64

Question 9.
a. 7⁶ ÷ 7³ = ____
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 ÷ 7³  =  7⁹
By simplification,
= 40353607

b. 4⁸ × 4³ = ____
Answer: 4194304
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.
4⁸ × 4³ = 4¹¹
By simplification,
= 4194304

c. 9⁵ × 9⁶ = ____
Answer: 31381059609
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.
9⁵ × 9⁶ = 9¹¹
By simplification,
= 31381059609

Question 10.
a. 2⁹ ÷ 2^-3 = ____
Answer: 0.000244
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.
2⁹ ÷ 2^-3 = 2¹²
By simplification,
= \(\frac{1}{3^{10}}\)
= \(\frac{1}{4096}\)
= 0.000244

b. 3⁸ ÷ 3² = ____
Answer: 59049
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.
3⁸ ÷ 3² =  3¹⁰
By simplification,
= 59049

c. 12⁴ × 12¹⁰ = ____
Answer: 1,283,918,464,548,864
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.
12⁴ × 12¹⁰ = 12¹⁴
By simplification,
= 1,283,918,464,548,864

Question 11.
a. 5⁴ × 5² = ____
Answer: 15,625
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.
5⁴ × 5² = 5⁶
By simplification,
= 15,625

b. 10⁷ ÷ 10⁴ = ____
Answer: 100000000000
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.
10⁷ ÷ 10⁴ =  10¹¹
By simplification,
=100000000000

c. 11³ × 11⁴ = ____
Answer: 19,487,171
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.
11³ × 11⁴ = 11⁷
By simplification,
= 19,487,171

Question 12.
a. 7⁵ ÷ 7² = ____
Answer: 823543
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.
7⁵ ÷ 7² =  7⁷
By simplification,
=823543

b. 6⁶ × 6³ = ____
Answer: 10,077,696
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.
6⁶ × 6³ = 6⁹
By simplification,
= 10,077,696

c. 12⁴ ÷ 12² = ____
Answer: 2985984
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.
12⁴ ÷ 12² =  12⁶
By simplification,
=2985984

Rewrite each in standard notation.

Question 13.
a. 9.545 × 10³
Answer: 9545
Translate numbers written in scientific notation into standard form by reading the exponent.
Move the decimal right 3 times
9.545 × 10³ = 9545

b. 8.596 × 10^-3
Answer: 0.008596
Translate numbers written in scientific notation into standard form by reading the exponent.
Move the decimal left 3 times
8.596 × 10^-3 = 0.008596

c. 9.318 × 10^-3
Answer: 0.009318
Translate numbers written in scientific notation into standard form by reading the exponent.
Move the decimal left 3 times
9.318 × 10^-3= 0.009318

Question 14.
a. 8.124 × 10⁶
Answer: 8124000
Translate numbers written in scientific notation into standard form by reading the exponent.
Move the decimal right 6 times
8.124 × 10⁶ = 8124000

b. 8.743 × 10⁴
Answer: 87430
Translate numbers written in scientific notation into standard form by reading the exponent.
Move the decimal right 4 times
8.743 × 10⁴ = 87430

c. 2.961 × 10⁵
Answer: 296100
Translate numbers written in scientific notation into standard form by reading the exponent.
Move the decimal right 5 times
2.961 × 10⁵ = 296100

Question 15.
a. 1.0428 × 10⁴
Answer: 10428
Translate numbers written in scientific notation into standard form by reading the exponent.
Move the decimal right 4 times
1.0428 × 10⁴ = 10428

b. 7.8543 × 10^-2
Answer: 0.078543
Translate numbers written in scientific notation into standard form by reading the exponent.
Move the decimal left 2 times
7.8543 × 10^-2 = 0.078543

c. 4.937 × 10^-4
Answer: 0.0004937
Translate numbers written in scientific notation into standard form by reading the exponent.
Move the decimal left 4 times
4.937 × 10^-4 = 0.0004937

Question 16.
a. 2.396 × 10⁵
Answer: 239600
Translate numbers written in scientific notation into standard form by reading the exponent.
Move the decimal right 5 times
2.396 × 10⁵= 239600

b. 8.352 × 10^-6
Answer: 0.000008352
Translate numbers written in scientific notation into standard form by reading the exponent.
Move the decimal left 6 times
8.352 × 10^-6 = 0.000008352

c. 3.85 × 10⁷
Answer: 38500000
Translate numbers written in scientific notation into standard form by reading the exponent.
Move the decimal right 7 times
3.85 × 10⁷= 38500000

Question 17.
a. 3.957 × 10²
Answer: 395.7
Translate numbers written in scientific notation into standard form by reading the exponent.
Move the decimal right 2 times
3.957 × 10²= 395.7

b. 9.389 × 10⁶
Answer: 9389000
Translate numbers written in scientific notation into standard form by reading the exponent.
Move the decimal right 6 times
9.389 × 10⁶= 9389000

b. 8.743 × 10⁴
Answer: 87430
Translate numbers written in scientific notation into standard form by reading the exponent.
Move the decimal right 4 times
8.743 × 10⁴= 87430

c. 4.109 × 10^-5
Answer: 0.00004109
Translate numbers written in scientific notation into standard form by reading the exponent.
Move the decimal left 5 times
4.109 × 10^-5 = 0.00004109

Rewrite each in scientific notation.

Question 18.
a. 0.4537
________
Answer: 4.537× 10^-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 right 1 places and multiply the shift by the power of 10.
0.4537 = 4.537× 10^-1

b. 0.006686
________
Answer: 6.686 × 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.
0.006686 = 6.686 × 10^-3

c. 133,300
________
Answer: 1.33300× 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.
133,300=1.33300× 10⁵

Question 19.
a. 0.7614
________
Answer: 7.614 × 10^-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 right 1 places and multiply the shift by the power of 10.
0.7614 = 7.614 × 10^-1

b. 0.01087
________
Answer: 1.087 × 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.
0.01087 = 1.087 × 10^-2

c. 517,700
________
Answer: 5.17700× 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.
517,700= 5.17700× 10⁵

Question 20.
a. 892,320
________
Answer: 8.92320× 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.
892,320=8.92320× 10⁵

b. 428,200
________
Answer: 4.28200× 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.
428,200=4.28200× 10⁵

c. 0.01283
________
Answer: 1.283 × 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.
0.01283 = 1.283 × 10^-2

Question 21.
a. 783,000
________
Answer: 7.83000× 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.
783,000 = 7.83000× 10⁵

b. 0.0004642
________
Answer: 4.642 × 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.
0.0004642 = 4.642 × 10^-4

c. 478,200,000
________
Answer: 4.78200000× 10⁸
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 8 places and multiply the shift by the power of 10.
478,200,000 = 4.78200000× 10⁸

Question 22.
a. 53,890,000
________
Answer: 5.3890000× 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.
53,890,000 = 5.3890000× 10⁷

b. 4,183,200,000
________
Answer: 4.183200000× 10⁹
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 9 places and multiply the shift by the power of 10.
4,183,200,000= 4.183200000× 10⁹

c. 0.00028737
________
Answer: 2.8737× 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.
0.00028737= 2.8737× 10^-4