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Exponents, Roots, Factorization of Whole Numbers: Proficiency Exam

Module by: Wade Ellis, Denny Burzynski. E-mail the authorsEdited By: Math Editors

Summary: This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module is a proficiency exam to the chapter "Exponents, Roots, Factorization of Whole Numbers." Each problem is accompanied with a reference link pointing back to the module that discusses the type of problem demonstrated in the question. The problems in this exam are accompanied by solutions.

Proficiency Exam

Exercise 1

((Reference)) In the number 8585, write the names used for the number 8 and the number 5.

Solution

base; exponent

Exercise 2

((Reference)) Write using exponents. 12×12×12×12×12×12×1212×12×12×12×12×12×12 size 12{"12" times "12" times "12" times "12" times "12" times "12" times "12"} {}

Solution

127127 size 12{"12" rSup { size 8{7} } } {}

Exercise 3

((Reference)) Expand 9494 size 12{9 rSup { size 8{4} } } {}.

Solution

94=9⋅9⋅9⋅9=6,56194=9⋅9⋅9⋅9=6,561 size 12{9 rSup { size 8{4} } =9 cdot 9 cdot 9 cdot 9=6,"561"} {}

For problems 4-15, determine the value of each expression.

Exercise 4

((Reference)) 4343 size 12{4 rSup { size 8{3} } } {}

Solution

64

Exercise 5

((Reference)) 1515 size 12{1 rSup { size 8{5} } } {}

Solution

1

Exercise 6

((Reference)) 0303 size 12{0 rSup { size 8{3} } } {}

Solution

0

Exercise 7

((Reference)) 2626 size 12{2 rSup { size 8{6} } } {}

Solution

64

Exercise 8

((Reference)) 4949 size 12{ sqrt {"49"} } {}

Solution

7

Exercise 9

((Reference)) 273273 size 12{ nroot { size 8{3} } {"27"} } {}

Solution

3

Exercise 10

((Reference)) 1818 size 12{ nroot { size 8{8} } {1} } {}

Solution

1

Exercise 11

((Reference)) 16+2⋅(8−6)16+2⋅(8−6) size 12{"16"+2 cdot \( 8 - 6 \) } {}

Solution

20

Exercise 12

((Reference)) 53−100+8⋅2−20÷553−100+8⋅2−20÷5 size 12{5 rSup { size 8{3} } - sqrt {"100"} +8 cdot 2 - "20" div 5} {}

Solution

127

Exercise 13

((Reference)) 3⋅82−2⋅3252−2⋅63−4⋅52293⋅82−2⋅3252−2⋅63−4⋅5229 size 12{3 cdot { {8 rSup { size 8{2} } - 2 cdot 3 rSup { size 8{2} } } over {5 rSup { size 8{2} } - 2} } cdot { {6 rSup { size 8{3} } - 4 cdot 5 rSup { size 8{2} } } over {"29"} } } {}

Solution

24

Exercise 14

((Reference)) 20+2423⋅2−5⋅2⋅5⋅7-81 7+3⋅220+2423⋅2−5⋅2⋅5⋅7-81 7+3⋅2 size 12{ { {"20"+2 rSup { size 8{4} } } over {2 rSup { size 8{3} } cdot 2 - 5 cdot 2} } cdot { {6 rSup { size 8{3} } - 4 cdot 5 rSup { size 8{2} } } over {"29"} } } {}

Solution

8

Exercise 15

((Reference)) 8−32+33−449−210−32+9−58−32+33−449−210−32+9−5 size 12{ left [ left (8 - 3 right ) rSup { size 8{2} } + left ("33" - 4 sqrt {"49"} right ) right ] - 2 left [ left ("10" - 3 rSup { size 8{2} } right )+9 right ] - 5} {}

Solution

5

For problems 16-20, find the prime factorization of each whole number. If the number is prime, write "prime."

Exercise 16

Solution

32⋅232⋅2 size 12{3 rSup { size 8{2} } cdot 2} {}

Exercise 17

Solution

22⋅1722⋅17 size 12{2 rSup { size 8{2} } cdot "17"} {}

Exercise 18

((Reference)) 142

Solution

2⋅712⋅71 size 12{2 cdot "71"} {}

Exercise 19

((Reference)) 151

Solution

prime

Exercise 20

((Reference)) 468

Solution

22⋅32⋅1322⋅32⋅13 size 12{2 rSup { size 8{2} } cdot 3"" lSup { size 8{2} } cdot "13"} {}

For problems 21 and 22, find the greatest common factor.

Exercise 21

((Reference)) 200 and 36

Solution

4

Exercise 22

((Reference)) 900 and 135

Solution

45

Exercise 23

((Reference)) Write all the factors of 36.

Solution

1, 2, 3, 4, 6, 9, 12, 18, 36

Exercise 24

((Reference)) Write all the divisors of 18.

Solution

1, 2, 3, 6, 9, 18

Exercise 25

((Reference)) Does 7 divide into 52⋅63⋅74⋅852⋅63⋅74⋅8 size 12{5 rSup { size 8{2} } cdot 6 rSup { size 8{3} } cdot 7 rSup { size 8{4} } cdot 8} {}? Explain.

Solution

Yes, because one of the (prime) factors of the number is 7.

Exercise 26

((Reference)) Is 3 a factor of 26⋅32⋅53⋅4626⋅32⋅53⋅46 size 12{2 rSup { size 8{6} } cdot 3 rSup { size 8{2} } cdot 5 rSup { size 8{3} } cdot 4 rSup { size 8{6} } } {}? Explain.

Solution

Yes, because it is one of the factors of the number.

Exercise 27

((Reference)) Does 13 divide into 113⋅124⋅152113⋅124⋅152 size 12{"11" rSup { size 8{3} } cdot "12" rSup { size 8{4} } cdot "15" rSup { size 8{2} } } {}? Explain.

Solution

No, because the prime 13 is not a factor any of the listed factors of the number.

For problems 28 and 29, find the least common multiple.

Exercise 28

((Reference)) 432 and 180

Solution

2,160

Exercise 29

((Reference)) 28, 40, and 95

Solution

5,320

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