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Exponents and Roots

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 discusses exponents and roots. By the end of the module students should be able to understand and be able to read exponential notation, understand the concept of root and be able to read root notation, and use a calculator having the yxyx key to determine a root.

Section Overview

  • Exponential Notation
  • Reading Exponential Notation
  • Roots
  • Reading Root Notation
  • Calculators

Exponential Notation

Exponential Notation

We have noted that multiplication is a description of repeated addition. Exponen­tial notation is a description of repeated multiplication.

Suppose we have the repeated multiplication

8 ⋅ 8 ⋅ 8 ⋅ 8 ⋅ 8 8 ⋅ 8 ⋅ 8 ⋅ 8 ⋅ 8 size 12{8 cdot 8 cdot 8 cdot 8 cdot 8} {}

Exponent

The factor 8 is repeated 5 times. Exponential notation uses a superscript for the number of times the factor is repeated. The superscript is placed on the repeated factor, 8585, in this case. The superscript is called an exponent.

The Function of an Exponent

An exponent records the number of identical factors that are repeated in a multiplication.

Sample Set A

Write the following multiplication using exponents.

Example 1

3⋅33⋅3 size 12{3 cdot 3} {}. Since the factor 3 appears 2 times, we record this as

3232 size 12{3 rSup { size 8{2} } } {}

Example 2

62⋅62⋅62⋅62⋅62⋅62⋅62⋅62⋅6262⋅62⋅62⋅62⋅62⋅62⋅62⋅62⋅62 size 12{"62" cdot "62" cdot "62" cdot "62" cdot "62" cdot "62" cdot "62" cdot "62" cdot "62"} {}. Since the factor 62 appears 9 times, we record this as

629629 size 12{"62" rSup { size 8{9} } } {}

Expand (write without exponents) each number.

Example 3

124124 size 12{"12" rSup { size 8{4} } } {}. The exponent 4 is recording 4 factors of 12 in a multiplication. Thus,

124=12⋅12⋅12⋅12124=12⋅12⋅12⋅12 size 12{"12" rSup { size 8{4} } ="12" cdot "12" cdot "12" cdot "12"} {}

Example 4

70637063 size 12{"706" rSup { size 8{3} } } {}. The exponent 3 is recording 3 factors of 706 in a multiplication. Thus,

7063=706⋅706⋅7067063=706⋅706⋅706 size 12{"706" rSup { size 8{3} } ="706" cdot "706" cdot "706"} {}

Practice Set A

Write the following using exponents.

Exercise 1

37⋅3737⋅37 size 12{"37" cdot "37"} {}

Solution

372372 size 12{"37" rSup { size 8{2} } } {}

Exercise 2

16⋅16⋅16⋅16⋅1616⋅16⋅16⋅16⋅16 size 12{"16" cdot "16" cdot "16" cdot "16" cdot "16"} {}

Solution

165165 size 12{"16" rSup { size 8{5} } } {}

Exercise 3

9⋅9⋅9⋅9⋅9⋅9⋅9⋅9⋅9⋅99⋅9⋅9⋅9⋅9⋅9⋅9⋅9⋅9⋅9 size 12{9 cdot 9 cdot 9 cdot 9 cdot 9 cdot 9 cdot 9 cdot 9 cdot 9 cdot 9} {}

Solution

910910 size 12{9 rSup { size 8{"10"} } } {}

Write each number without exponents.

Exercise 4

853853 size 12{"85" rSup { size 8{3} } } {}

Solution

85⋅85⋅8585⋅85⋅85 size 12{"85" cdot "85" cdot "85"} {}

Exercise 5

4747 size 12{4 rSup { size 8{7} } } {}

Solution

4⋅4⋅4⋅4⋅4⋅4⋅44⋅4⋅4⋅4⋅4⋅4⋅4 size 12{4 cdot 4 cdot 4 cdot 4 cdot 4 cdot 4 cdot 4} {}

Exercise 6

1,73921,7392 size 12{1,"739" rSup { size 8{2} } } {}

Solution

1,739⋅1,7391,739⋅1,739 size 12{1,"739" cdot 1,"739"} {}

Reading Exponential Notation

In a number such as 8585 size 12{8 rSup { size 8{5} } } {},

Base

8 is called the base.

Exponent, Power

5 is called the exponent, or power. 8585 size 12{8 rSup { size 8{5} } } {} is read as "eight to the fifth power," or more simply as "eight to the fifth," or "the fifth power of eight."

Squared

When a whole number is raised to the second power, it is said to be squared. The number 5252 size 12{5 rSup { size 8{2} } } {} can be read as

5 to the second power, or
5 to the second, or
5 squared.

Cubed

When a whole number is raised to the third power, it is said to be cubed. The number 5353 size 12{5 rSup { size 8{3} } } {} can be read as

5 to the third power, or
5 to the third, or
5 cubed.

When a whole number is raised to the power of 4 or higher, we simply say that that number is raised to that particular power. The number 5858 size 12{5 rSup { size 8{8} } } {} can be read as

5 to the eighth power, or just
5 to the eighth.

Roots

In the English language, the word "root" can mean a source of something. In mathematical terms, the word "root" is used to indicate that one number is the source of another number through repeated multiplication.

Square Root

We know that 49=7249=72 size 12{"49"=7 rSup { size 8{2} } } {}, that is, 49=7⋅749=7⋅7 size 12{"49"=7 cdot 7} {}. Through repeated multiplication, 7 is the source of 49. Thus, 7 is a root of 49. Since two 7's must be multiplied together to produce 49, the 7 is called the second or square root of 49.

Cube Root

We know that 8=238=23 size 12{8=2 rSup { size 8{3} } } {}, that is, 8=2⋅2⋅28=2⋅2⋅2 size 12{8=2 cdot 2 cdot 2} {}. Through repeated multiplication, 2 is the source of 8. Thus, 2 is a root of 8. Since three 2's must be multiplied together to produce 8, 2 is called the third or cube root of 8.

We can continue this way to see such roots as fourth roots, fifth roots, sixth roots, and so on.

Reading Root Notation

There is a symbol used to indicate roots of a number. It is called the radical sign nn

The Radical Sign nn

The symbol nn is called a radical sign and indicates the nth root of a number.

We discuss particular roots using the radical sign as follows:

Square Root

number2number2 indicates the square root of the number under the radical sign. It is customary to drop the 2 in the radical sign when discussing square roots. The symbol is understood to be the square root radical sign.

4949 size 12{ sqrt {"49"} } {} = 7 since 7⋅7=72=497⋅7=72=49 size 12{7 cdot 7=7 rSup { size 8{2} } ="49"} {}

Cube Root

number3number3 size 12{ nroot { size 8{3} } { ital "number"} } {} indicates the cube root of the number under the radical sign.

83=283=2 size 12{ nroot { size 8{3} } {8} =2} {} since 2⋅2⋅2=23=82⋅2⋅2=23=8 size 12{2 cdot 2 cdot 2=2 rSup { size 8{3} } =8} {}

Fourth Root

number4number4 size 12{ nroot { size 8{4} } { ital "number"} } {} indicates the fourth root of the number under the radical sign.

814=3814=3 size 12{ nroot { size 8{4} } {"81"} =3} {} since 3⋅3⋅3⋅3=34=813⋅3⋅3⋅3=34=81 size 12{3 cdot 3 cdot 3 cdot 3=3 rSup { size 8{4} } ="81"} {}

In an expression such as 325325 size 12{ nroot { size 8{5} } {"32"} } {}

Radical Sign

is called the radical sign.

Index

5 is called the index. (The index describes the indicated root.)

Radicand

32 is called the radicand.

Radical

325325 size 12{ nroot { size 8{5} } {"32"} } {} is called a radical (or radical expression).

Sample Set B

Find each root.

Example 5

2525 size 12{ sqrt {"25"} } {} To determine the square root of 25, we ask, "What whole number squared equals 25?" From our experience with multiplication, we know this number to be 5. Thus,

25 = 5 25 = 5 size 12{ sqrt {"25"} =5} {}

Check: 5 ⋅ 5 = 5 2 = 25 5 ⋅ 5 = 5 2 = 25 size 12{5 cdot 5=5 rSup { size 8{2} } ="25"} {}

Example 6

325325 size 12{ nroot { size 8{5} } {"32"} } {} To determine the fifth root of 32, we ask, "What whole number raised to the fifth power equals 32?" This number is 2.

32 5 = 2 32 5 = 2 size 12{ nroot { size 8{5} } {"32"} =2} {}

Check: 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 = 2 5 = 32 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 = 2 5 = 32 size 12{2 cdot 2 cdot 2 cdot 2 cdot 2=2 rSup { size 8{5} } ="32"} {}

Practice Set B

Find the following roots using only a knowledge of multiplication.

Exercise 7

6464 size 12{ sqrt {"64"} } {}

Solution

8

Exercise 8

100100 size 12{ sqrt {"100"} } {}

Solution

10

Exercise 9

643643 size 12{ nroot { size 8{3} } {"64"} } {}

Solution

4

Exercise 10

646646 size 12{ nroot { size 8{6} } {"64"} } {}

Solution

2

Calculators

Calculators with the xx size 12{ sqrt {x} } {}, yxyx size 12{y rSup { size 8{x} } } {}, and 1/x1/x size 12{1/x} {} keys can be used to find or approximate roots.

Sample Set C

Example 7

Use the calculator to find 121121 size 12{ sqrt {"121"} } {}

Table 1
    Display Reads
Type 121 121
Press x x size 12{ sqrt {x} } {} 11

Example 8

Find 2187721877.

Table 2
    Display Reads
Type 2187 2187
Press y x y x size 12{y rSup { size 8{x} } } {} 2187
Type 7 7
Press 1 / x 1 / x size 12{1/x} {} .14285714
Press = 3

21877=321877=3 (Which means that 3 7 = 2187 3 7 =2187 .)

Practice Set C

Use a calculator to find the following roots.

Exercise 11

72937293 size 12{ nroot { size 8{3} } {"729"} } {}

Solution

9

Exercise 12

8503056485030564 size 12{ nroot { size 8{4} } {"8503056"} } {}

Solution

54

Exercise 13

5336153361 size 12{ sqrt {"53361"} } {}

Solution

231

Exercise 14

16777216121677721612 size 12{ nroot { size 8{"12"} } {"16777216"} } {}

Solution

4

Exercises

For the following problems, write the expressions using expo­nential notation.

Exercise 15

4 ⋅ 4 4 ⋅ 4 size 12{4 cdot 4} {}

Solution

4 2 4 2 size 12{4 rSup { size 8{2} } } {}

Exercise 16

12⋅1212⋅12 size 12{"12" cdot "12"} {}

Exercise 17

9 ⋅ 9 ⋅ 9 ⋅ 9 9 ⋅ 9 ⋅ 9 ⋅ 9 size 12{9 cdot 9 cdot 9 cdot 9} {}

Solution

9 4 9 4 size 12{9 rSup { size 8{4} } } {}

Exercise 18

10⋅10⋅10⋅10⋅10⋅1010⋅10⋅10⋅10⋅10⋅10 size 12{"10" cdot "10" cdot "10" cdot "10" cdot "10" cdot "10"} {}

Exercise 19

826 ⋅ 826 ⋅ 826 826 ⋅ 826 ⋅ 826 size 12{"826" cdot "826" cdot "826"} {}

Solution

826 3 826 3 size 12{"826" rSup { size 8{3} } } {}

Exercise 20

3,021⋅3,021⋅3,021⋅3,021⋅3,0213,021⋅3,021⋅3,021⋅3,021⋅3,021 size 12{3,"021" cdot 3,"021" cdot 3,"021" cdot 3,"021" cdot 3,"021"} {}

Exercise 21

6 · 6 ····· 6 ︸ 85 factors of 6 6 · 6 ····· 6 ︸ 85 factors of 6

Solution

685685 size 12{6 rSup { size 8{"85"} } } {}

Exercise 22

2 · 2 ····· 2 ︸ 112 factors of 2 2 · 2 ····· 2 ︸ 112 factors of 2

Exercise 23

1 · 1 ····· 1 ︸ 3,008 factors of 1 1 · 1 ····· 1 ︸ 3,008 factors of 1

Solution

1300813008 size 12{1 rSup { size 8{"3008"} } } {}

For the following problems, expand the terms. (Do not find the actual value.)

Exercise 24

5353 size 12{5 rSup { size 8{3} } } {}

Exercise 25

7 4 7 4 size 12{7 rSup { size 8{4} } } {}

Solution

7 ⋅ 7 ⋅ 7 ⋅ 7 7 ⋅ 7 ⋅ 7 ⋅ 7 size 12{7 cdot 7 cdot 7 cdot 7} {}

Exercise 26

152152 size 12{"15" rSup { size 8{2} } } {}

Exercise 27

117 5 117 5 size 12{"117" rSup { size 8{5} } } {}

Solution

117 ⋅ 117 ⋅ 117 ⋅ 117 ⋅ 117 117 ⋅ 117 ⋅ 117 ⋅ 117 ⋅ 117 size 12{"117" cdot "117" cdot "117" cdot "117" cdot "117"} {}

Exercise 28

616616 size 12{"61" rSup { size 8{6} } } {}

Exercise 29

30 2 30 2 size 12{"30" rSup { size 8{2} } } {}

Solution

30 ⋅ 30 30 ⋅ 30 size 12{"30" cdot "30"} {}

For the following problems, determine the value of each of the powers. Use a calculator to check each result.

Exercise 30

3232 size 12{3 rSup { size 8{2} } } {}

Exercise 31

4242 size 12{4 rSup { size 8{2} } } {}

Solution

4⋅4=164⋅4=16 size 12{4 cdot 4="16"} {}

Exercise 32

1212 size 12{1 rSup { size 8{2} } } {}

Exercise 33

102102 size 12{"10" rSup { size 8{2} } } {}

Solution

10⋅10=10010⋅10=100 size 12{"10" cdot "10"="100"} {}

Exercise 34

112112 size 12{"11" rSup { size 8{2} } } {}

Exercise 35

122122 size 12{"12" rSup { size 8{2} } } {}

Solution

12⋅12=14412⋅12=144 size 12{"12" cdot "12"="144"} {}

Exercise 36

132132 size 12{"13" rSup { size 8{2} } } {}

Exercise 37

152152 size 12{"15" rSup { size 8{2} } } {}

Solution

15⋅15=22515⋅15=225 size 12{"15" cdot "15"="225"} {}

Exercise 38

1414 size 12{1 rSup { size 8{4} } } {}

Exercise 39

3434 size 12{3 rSup { size 8{4} } } {}

Solution

3⋅3⋅3⋅3=813⋅3⋅3⋅3=81 size 12{3 cdot 3 cdot 3 cdot 3 ="81"} {}

Exercise 40

7373 size 12{7 rSup { size 8{3} } } {}

Exercise 41

103103 size 12{"10" rSup { size 8{3} } } {}

Solution

10⋅10⋅10=1,00010⋅10⋅10=1,000 size 12{"10" cdot "10" cdot "10"=1,"000"} {}

Exercise 42

10021002 size 12{"100" rSup { size 8{2} } } {}

Exercise 43

8383 size 12{8 rSup { size 8{3} } } {}

Solution

8⋅8⋅8=5128⋅8⋅8=512 size 12{8 cdot 8 cdot 8="512"} {}

Exercise 44

5555 size 12{5 rSup { size 8{5} } } {}

Exercise 45

9393 size 12{9 rSup { size 8{3} } } {}

Solution

9⋅9⋅9=7299⋅9⋅9=729 size 12{9 cdot 9 cdot 9="729"} {}

Exercise 46

6262 size 12{6 rSup { size 8{2} } } {}

Exercise 47

7171 size 12{7 rSup { size 8{1} } } {}

Solution

71=771=7 size 12{7 rSup { size 8{1} } =7} {}

Exercise 48

128128 size 12{1 rSup { size 8{"28"} } } {}

Exercise 49

2727 size 12{2 rSup { size 8{7} } } {}

Solution

2⋅2⋅2⋅2⋅2⋅2⋅2=1282⋅2⋅2⋅2⋅2⋅2⋅2=128 size 12{2 cdot 2 cdot 2 cdot 2 cdot 2 cdot 2 cdot 2="128"} {}

Exercise 50

0505 size 12{0 rSup { size 8{5} } } {}

Exercise 51

8484 size 12{8 rSup { size 8{4} } } {}

Solution

8⋅8⋅8⋅8=4,0968⋅8⋅8⋅8=4,096 size 12{8 cdot 8 cdot 8 cdot 8=4,"096"} {}

Exercise 52

5858 size 12{5 rSup { size 8{8} } } {}

Exercise 53

6969 size 12{6 rSup { size 8{9} } } {}

Solution

6⋅6⋅6⋅6⋅6⋅6⋅6⋅6⋅6=10,077,6966⋅6⋅6⋅6⋅6⋅6⋅6⋅6⋅6=10,077,696 size 12{6 cdot 6 cdot 6 cdot 6 cdot 6 cdot 6 cdot 6 cdot 6 cdot 6="10","077","696"} {}

Exercise 54

253253 size 12{"25" rSup { size 8{3} } } {}

Exercise 55

422422 size 12{"42" rSup { size 8{2} } } {}

Solution

42⋅42=1,76442⋅42=1,764 size 12{"42" cdot "42"=1,"764"} {}

Exercise 56

313313 size 12{"31" rSup { size 8{3} } } {}

Exercise 57

155155 size 12{"15" rSup { size 8{5} } } {}

Solution

15⋅15⋅15⋅15⋅15=759,37515⋅15⋅15⋅15⋅15=759,375 size 12{"15" cdot "15" cdot "15" cdot "15" cdot "15"="759","375"} {}

Exercise 58

220220 size 12{2 rSup { size 8{"20"} } } {}

Exercise 59

81628162 size 12{"816" rSup { size 8{2} } } {}

Solution

816⋅816=665,856816⋅816=665,856 size 12{"816" cdot "816"="665","856"} {}

For the following problems, find the roots (using your knowledge of multiplication). Use a calculator to check each result.

Exercise 60

99 size 12{ sqrt {9} } {}

Exercise 61

1616 size 12{ sqrt {"16"} } {}

Solution

4

Exercise 62

3636 size 12{ sqrt {"36"} } {}

Exercise 63

6464 size 12{ sqrt {"64"} } {}

Solution

8

Exercise 64

121121 size 12{ sqrt {"121"} } {}

Exercise 65

144144 size 12{ sqrt {"144"} } {}

Solution

12

Exercise 66

169169 size 12{ sqrt {"169"} } {}

Exercise 67

225225 size 12{ sqrt {"225"} } {}

Solution

15

Exercise 68

273273 size 12{ nroot { size 8{3} } {"27"} } {}

Exercise 69

325325 size 12{ nroot { size 8{5} } {"32"} } {}

Solution

2

Exercise 70

25642564 size 12{ nroot { size 8{4} } {"256"} } {}

Exercise 71

21632163 size 12{ nroot { size 8{3} } {"216"} } {}

Solution

6

Exercise 72

1717 size 12{ nroot { size 8{7} } {1} } {}

Exercise 73

400400 size 12{ sqrt {"400"} } {}

Solution

20

Exercise 74

900900 size 12{ sqrt {"900"} } {}

Exercise 75

10,00010,000 size 12{ sqrt {"10","000"} } {}

Solution

100

Exercise 76

324324 size 12{ sqrt {"324"} } {}

Exercise 77

3,6003,600 size 12{ sqrt {3,"600"} } {}

Solution

60

For the following problems, use a calculator with the keys xx size 12{ sqrt {x} } {}, yxyx size 12{y rSup { size 8{x} } } {}, and 1/x1/x size 12{1/x} {} to find each of the values.

Exercise 78

676676 size 12{ sqrt {"676"} } {}

Exercise 79

1,1561,156 size 12{ sqrt {1,"156"} } {}

Solution

34

Exercise 80

46,22546,225 size 12{ sqrt {"46","225"} } {}

Exercise 81

17,288,96417,288,964 size 12{ sqrt {"17","288","964"} } {}

Solution

4,158

Exercise 82

3,37533,3753 size 12{ nroot { size 8{3} } {3,"375"} } {}

Exercise 83

331,7764331,7764 size 12{ nroot { size 8{4} } {"331","776"} } {}

Solution

24

Exercise 84

5,764,80185,764,8018 size 12{ nroot { size 8{8} } {5,"764","801"} } {}

Exercise 85

16,777,2161216,777,21612 size 12{ nroot { size 8{"12"} } {"16","777","216"} } {}

Solution

4

Exercise 86

16,777,216816,777,2168 size 12{ nroot { size 8{8} } {"16","777","216"} } {}

Exercise 87

9,765,625109,765,62510 size 12{ nroot { size 8{"10"} } {9,"765","625"} } {}

Solution

5

Exercise 88

160,0004160,0004 size 12{ nroot { size 8{4} } {"160","000"} } {}

Exercise 89

531,4413531,4413 size 12{ nroot { size 8{3} } {"531","441"} } {}

Solution

81

Exercises for Review

Exercise 90

((Reference)) Use the numbers 3, 8, and 9 to illustrate the associative property of addition.

Exercise 91

((Reference)) In the multiplication 8⋅4=328⋅4=32 size 12{8 cdot 4="32"} {}, specify the name given to the num­bers 8 and 4.

Solution

8 is the multiplier; 4 is the multiplicand

Exercise 92

((Reference)) Does the quotient 15÷015÷0 exist? If so, what is it?

Exercise 93

((Reference)) Does the quotient 0÷150÷15exist? If so, what is it?

Solution

Yes; 0

Exercise 94

((Reference)) Use the numbers 4 and 7 to illustrate the commutative property of multiplication.

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