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Exponents

Module by: Wade Ellis, Denny Burzynski. E-mail the authors

Summary: This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. The symbols, notations, and properties of numbers that form the basis of algebra, as well as exponents and the rules of exponents, are introduced in this chapter. Each property of real numbers and the rules of exponents are expressed both symbolically and literally. Literal explanations are included because symbolic explanations alone may be difficult for a student to interpret. Objectives of this module: understand exponential notation, be able to read exponential notation, understand how to use exponential notation with the order of operations.

Overview

  • Exponential Notation
  • Reading Exponential Notation
  • The Order of Operations

Exponential Notation

In Section (Reference) we were reminded that multiplication is a description for repeated addition. A natural question is “Is there a description for repeated multiplication?” The answer is yes. The notation that describes repeated multiplication is exponential notation.

Factors

In multiplication, the numbers being multiplied together are called factors. In repeated multiplication, all the factors are the same. In nonrepeated multiplication, none of the factors are the same. For example,

Example 1

18⋅18⋅18⋅18 Repeated multiplication of 18. All four factors, 18, are the same. x⋅x⋅x⋅x⋅x Repeated multiplication of x. All five factors, x,are the same. 3⋅7⋅a Nonrepeated multiplication. None of the factors are the same. 18⋅18⋅18⋅18 Repeated multiplication of 18. All four factors, 18, are the same. x⋅x⋅x⋅x⋅x Repeated multiplication of x. All five factors, x,are the same. 3⋅7⋅a Nonrepeated multiplication. None of the factors are the same.

Exponential notation is used to show repeated multiplication of the same factor. The notation consists of using a superscript on the factor that is repeated. The superscript is called an exponent.

Exponential Notation

If x x is any real number and n n is a natural number, then

 x n = x⋅x⋅x⋅...⋅x ︸ n factors of x x n = x⋅x⋅x⋅...⋅x ︸ n factors of x

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

Note that the definition for exponential notation only has meaning for natural number exponents. We will extend this notation to include other numbers as exponents later.

Sample Set A

Example 2

7⋅7⋅7⋅7⋅7⋅7= 7 6 7⋅7⋅7⋅7⋅7⋅7= 7 6 .

The repeated factor is 7. The exponent 6 records the fact that 7 appears 6 times in the multiplication.

Example 3

x⋅x⋅x⋅x= x 4 x⋅x⋅x⋅x= x 4 .

The repeated factor is x x . The exponent 4 records the fact that x x appears 4 times in the multiplication.

Example 4

(2y)(2y)(2y)= (2y) 3 (2y)(2y)(2y)= (2y) 3 .

The repeated factor is 2y 2y . The exponent 3 records the fact that the factor 2y 2y appears 3 times in the multiplication.

Example 5

2yyy=2 y 3 2yyy=2 y 3 .

The repeated factor is y y . The exponent 3 records the fact that the factor y y appears 3 times in the multiplication.

Example 6

(a+b)(a+b)(a−b)(a−b)(a−b)= (a+b) 2 (a−b) 3 (a+b)(a+b)(a−b)(a−b)(a−b)= (a+b) 2 (a−b) 3 .

The repeated factors are (a+b) (a+b) and (a−b) (a−b) , (a+b) (a+b) appearing 2 times and (a−b) (a−b) appearing 3 times.

Practice Set A

Write each of the following using exponents.

Exercise 1

a⋅a⋅a⋅a a⋅a⋅a⋅a

Solution

a 4 a 4

Exercise 2

(3b)(3b)(5c)(5c)(5c)(5c) (3b)(3b)(5c)(5c)(5c)(5c)

Solution

(3b) 2 (5c) 4 (3b) 2 (5c) 4

Exercise 3

2⋅2⋅7⋅7⋅7⋅(a−4)(a−4) 2⋅2⋅7⋅7⋅7⋅(a−4)(a−4)

Solution

2 2 ⋅ 7 3 (a−4) 2 2 2 ⋅ 7 3 (a−4) 2

Exercise 4

8xxxyzzzzz 8xxxyzzzzz

Solution

8 x 3 y z 5 8 x 3 y z 5

CAUTION

It is extremely important to realize and remember that an exponent applies only to the factor to which it is directly connected.

Sample Set B

Example 7

8 x 3 8 x 3 means 8⋅xxx 8⋅xxx and not 8x8x8x 8x8x8x . The exponent 3 applies only to the factor x x since it is only to the factor x x that the 3 is connected.

Example 8

(8x) 3 (8x) 3 means (8x)(8x)(8x) (8x)(8x)(8x) since the parentheses indicate that the exponent 3 is directly connected to the factor 8x 8x . Remember that the grouping symbols (   ) (   ) indicate that the quantities inside are to be considered as one single number.

Example 9

34 (a+1) 2 34 (a+1) 2 means 34⋅(a+1)(a+1) 34⋅(a+1)(a+1) since the exponent 2 applies only to the factor (a+1) (a+1) .

Practice Set B

Write each of the following without exponents.

Exercise 5

4 a 3 4 a 3

Solution

4aaa 4aaa

Exercise 6

(4a) 3 (4a) 3

Solution

(4a)(4a)(4a) (4a)(4a)(4a)

Sample Set C

Example 10

Select a number to show that (2x) 2 (2x) 2 is not always equal to 2 x 2 2 x 2 .

Suppose we choose x x to be 5. Consider both (2x) 2 (2x) 2 and 2 x 2 2 x 2 .

(2x) 2 2 x 2 (2⋅5) 2 2⋅ 5 2 (10) 2 2⋅25 100 ≠ 50 (2x) 2 2 x 2 (2⋅5) 2 2⋅ 5 2 (10) 2 2⋅25 100 ≠ 50
(1)

Notice that (2x) 2 =2 x 2 (2x) 2 =2 x 2 only when x=0 x=0 .

Practice Set C

Exercise 7

Select a number to show that (5x) 2 (5x) 2 is not always equal to 5 x 2 5 x 2 .

Solution

Select x=3 x=3 . Then (5⋅3) 2 = (15) 2 =225 (5⋅3) 2 = (15) 2 =225 , but 5⋅ 3 2 =5⋅9=45 5⋅ 3 2 =5⋅9=45 .     225≠45 225≠45 .

Reading Exponential Notation

In x n x n ,

Base

x x is the base

Exponent

n n is the exponent

Power

The number represented by x n x n is called a power.

xx to the nnth Power

The term x n x n is read as " x x to the n n th power," or more simply as " x x to the n n th."

xx Squared and xx Cubed

The symbol x 2 x 2 is often read as " x x squared," and x 3 x 3 is often read as " x x cubed." A natural question is "Why are geometric terms appearing in the exponent expression?" The answer for x 3 x 3 is this: x 3 x 3 means x⋅x⋅x x⋅x⋅x . In geometry, the volume of a rectangular box is found by multiplying the length by the width by the depth. A cube has the same length on each side. If we represent this length by the letter x x then the volume of the cube is x⋅x⋅x x⋅x⋅x , which, of course, is described by x 3 x 3 . (Can you think of why x 2 x 2 is read as x x squared?)

Cube with
length =x =x
width =x =x
depth =x =x
Volume =xxx= x 3 =xxx= x 3

A cube with length of side equal to x.

The Order of Operations

In Section (Reference) we were introduced to the order of operations. It was noted that we would insert another operation before multiplication and division. We can do that now.

The Order of Operations

  1. Perform all operations inside grouping symbols beginning with the innermost set.
  2. Perform all exponential operations as you come to them, moving left-to-right.
  3. Perform all multiplications and divisions as you come to them, moving left-to-right.
  4. Perform all additions and subtractions as you come to them, moving left-to-right.

Sample Set D

Use the order of operations to simplify each of the following.

Example 11

2 2 +5=4+5=9 2 2 +5=4+5=9

Example 12

5 2 + 3 2 +10=25+9+10=44 5 2 + 3 2 +10=25+9+10=44

Example 13

2 2 +(5)(8)−1 =4+(5)(8)−1 =4+40−1 =43 2 2 +(5)(8)−1 =4+(5)(8)−1 =4+40−1 =43

Example 14

7⋅6− 4 2 + 1 5 =7⋅6−16+1 =42−16+1 =27 7⋅6− 4 2 + 1 5 =7⋅6−16+1 =42−16+1 =27

Example 15

(2+3) 3 + 7 2 −3 (4+1) 2 = (5) 3 + 7 2 −3 (5) 2 =125+49−3(25) =125+49−75 =99 (2+3) 3 + 7 2 −3 (4+1) 2 = (5) 3 + 7 2 −3 (5) 2 =125+49−3(25) =125+49−75 =99

Example 16

[4 (6+2) 3 ] 2 = [4 (8) 3 ] 2 = [4(512)] 2 = [2048] 2 =4,194,304 [4 (6+2) 3 ] 2 = [4 (8) 3 ] 2 = [4(512)] 2 = [2048] 2 =4,194,304

Example 17

6( 3 2 + 2 2 )+ 4 2 =6(9+4)+ 4 2 =6(13)+ 4 2 =6(13)+16 =78+16 =94 6( 3 2 + 2 2 )+ 4 2 =6(9+4)+ 4 2 =6(13)+ 4 2 =6(13)+16 =78+16 =94

Example 18

6 2 + 2 2 4 2 +6⋅ 2 2 + 1 3 + 8 2 10 2 −(19)(5) = 36+4 16+6⋅4 + 1+64 100−95 = 36+4 16+24 + 1+64 100−95 = 40 40 + 65 5 =1+13 =14 6 2 + 2 2 4 2 +6⋅ 2 2 + 1 3 + 8 2 10 2 −(19)(5) = 36+4 16+6⋅4 + 1+64 100−95 = 36+4 16+24 + 1+64 100−95 = 40 40 + 65 5 =1+13 =14

Practice Set D

Use the order of operations to simplify the following.

Exercise 8

3 2 +4⋅5 3 2 +4⋅5

Solution

29

Exercise 9

2 3 + 3 3 −8⋅4 2 3 + 3 3 −8⋅4

Solution

3

Exercise 10

1 4 + ( 2 2 +4) 2 ÷ 2 3 1 4 + ( 2 2 +4) 2 ÷ 2 3

Solution

9

Exercise 11

[6(10− 2 3 )] 2 − 10 2 − 6 2 [6(10− 2 3 )] 2 − 10 2 − 6 2

Solution

8

Exercise 12

5 2 + 6 2 −10 1+ 4 2 + 0 4 − 0 5 7 2 −6⋅ 2 3 5 2 + 6 2 −10 1+ 4 2 + 0 4 − 0 5 7 2 −6⋅ 2 3

Solution

3

Exercises

For the following problems, write each of the quantities using exponential notation.

Exercise 13

b b to the fourth

Solution

b 4 b 4

Exercise 14

a a squared

Exercise 15

x x to the eighth

Solution

x 8 x 8

Exercise 16

(−3) (−3) cubed

Exercise 17

5 times s s squared

Solution

5 s 2 5 s 2

Exercise 18

3 squared times y y to the fifth

Exercise 19

a a cubed minus (b+7) (b+7) squared

Solution

a 3 − ( b+7 ) 2 a 3 − ( b+7 ) 2

Exercise 20

(21−x) (21−x) cubed plus (x+5) (x+5) to the seventh

Exercise 21

xxxxx xxxxx

Solution

x 5 x 5

Exercise 22

(8)(8)xxxx (8)(8)xxxx

Exercise 23

2⋅3⋅3⋅3⋅3xxyyyyy 2⋅3⋅3⋅3⋅3xxyyyyy

Solution

2( 3 4 ) x 2 y 5 2( 3 4 ) x 2 y 5

Exercise 24

2⋅2⋅5⋅6⋅6⋅6xyyzzzwwww 2⋅2⋅5⋅6⋅6⋅6xyyzzzwwww

Exercise 25

7xx(a+8)(a+8) 7xx(a+8)(a+8)

Solution

7 x 2 ( a+8 ) 2 7 x 2 ( a+8 ) 2

Exercise 26

10xyy(c+5)(c+5)(c+5) 10xyy(c+5)(c+5)(c+5)

Exercise 27

4x4x4x4x4x 4x4x4x4x4x

Solution

( 4x ) 5  or  4 5 x 5 ( 4x ) 5  or  4 5 x 5

Exercise 28

(9a)(9a)(9a)(9a) (9a)(9a)(9a)(9a)

Exercise 29

(−7)(−7)(−7)aabbba(−7)baab (−7)(−7)(−7)aabbba(−7)baab

Solution

( −7 ) 4 a 5 b 5 ( −7 ) 4 a 5 b 5

Exercise 30

(a−10)(a−10)(a+10) (a−10)(a−10)(a+10)

Exercise 31

(z+w)(z+w)(z+w)(z−w)(z−w) (z+w)(z+w)(z+w)(z−w)(z−w)

Solution

( z+w ) 3 ( z−w ) 2 ( z+w ) 3 ( z−w ) 2

Exercise 32

(2y)(2y)2y2y (2y)(2y)2y2y

Exercise 33

3xyxxy−(x+1)(x+1)(x+1) 3xyxxy−(x+1)(x+1)(x+1)

Solution

3 x 3 y 2 − ( x+1 ) 3 3 x 3 y 2 − ( x+1 ) 3

For the following problems, expand the quantities so that no exponents appear.

Exercise 34

4 3 4 3

Exercise 35

6 2 6 2

Solution

6 · 6 6 · 6

Exercise 36

7 3 y 2 7 3 y 2

Exercise 37

8 x 3 y 2 8 x 3 y 2

Solution

8 · x · x · x · y · y 8 · x · x · x · y · y

Exercise 38

(18 x 2 y 4 ) 2 (18 x 2 y 4 ) 2

Exercise 39

(9 a 3 b 2 ) 3 (9 a 3 b 2 ) 3

Solution

( 9aaabb )( 9aaabb )( 9aaabb ) or 9 · 9 · 9aaaaaaaaabbbbbb ( 9aaabb )( 9aaabb )( 9aaabb ) or 9 · 9 · 9aaaaaaaaabbbbbb

Exercise 40

5 x 2 (2 y 3 ) 3 5 x 2 (2 y 3 ) 3

Exercise 41

10 a 3 b 2 (3c) 2 10 a 3 b 2 (3c) 2

Solution

10aaabb( 3c )( 3c ) or 10 · 3 · 3aaabbcc 10aaabb( 3c )( 3c ) or 10 · 3 · 3aaabbcc

Exercise 42

(a+10) 2 ( a 2 +10) 2 (a+10) 2 ( a 2 +10) 2

Exercise 43

( x 2 − y 2 )( x 2 + y 2 ) ( x 2 − y 2 )( x 2 + y 2 )

Solution

( xx−yy )( xx+yy ) ( xx−yy )( xx+yy )

For the following problems, select a number (or numbers) to show that

Exercise 44

(5x) 2 (5x) 2 is not generally equal to 5 x 2 5 x 2 .

Exercise 45

(7x) 2 (7x) 2 is not generally equal to 7 x 2 7 x 2 .

Solution

Select x=2. x=2. Then, 196≠28. 196≠28.

Exercise 46

(a+b) 2 (a+b) 2 is not generally equal to a 2 + b 2 a 2 + b 2 .

Exercise 47

For what real number is (6a) 2 (6a) 2 equal to 6 a 2 6 a 2 ?

Solution

zero

Exercise 48

For what real numbers, a a and b b , is (a+b) 2 (a+b) 2 equal to a 2 + b 2 a 2 + b 2 ?

Use the order of operations to simplify the quantities for the following problems.

Exercise 49

3 2 +7 3 2 +7

Solution

16

Exercise 50

4 3 −18 4 3 −18

Exercise 51

5 2 +2(40) 5 2 +2(40)

Solution

105

Exercise 52

8 2 +3+5(2+7) 8 2 +3+5(2+7)

Exercise 53

2 5 +3(8+1) 2 5 +3(8+1)

Solution

59

Exercise 54

3 4 + 2 4 (1+5) 3 3 4 + 2 4 (1+5) 3

Exercise 55

( 6 2 − 4 2 )÷5 ( 6 2 − 4 2 )÷5

Solution

4

Exercise 56

2 2 (10− 2 3 ) 2 2 (10− 2 3 )

Exercise 57

( 3 4 − 4 3 )÷17 ( 3 4 − 4 3 )÷17

Solution

1

Exercise 58

(4+3) 2 +1÷(2⋅5) (4+3) 2 +1÷(2⋅5)

Exercise 59

( 2 4 + 2 5 − 2 3 ⋅5) 2 ÷ 4 2 ( 2 4 + 2 5 − 2 3 ⋅5) 2 ÷ 4 2

Solution

4

Exercise 60

1 6 + 0 8 + 5 2 (2+8) 3 1 6 + 0 8 + 5 2 (2+8) 3

Exercise 61

(7)(16)− 9 2 +4( 1 1 + 3 2 ) (7)(16)− 9 2 +4( 1 1 + 3 2 )

Solution

71

Exercise 62

2 3 −7 5 2 2 3 −7 5 2

Exercise 63

(1+6) 2 +2 19 (1+6) 2 +2 19

Solution

51 19 51 19

Exercise 64

6 2 −1 5 + 4 3 +(2)(3) 10 6 2 −1 5 + 4 3 +(2)(3) 10

Exercise 65

5[ 8 2 −9(6)] 2 5 −7 + 7 2 − 4 2 2 4 −5 5[ 8 2 −9(6)] 2 5 −7 + 7 2 − 4 2 2 4 −5

Solution

5

Exercise 66

(2+1) 3 + 2 3 + 1 3 6 2 − 15 2 − [2(5)] 2 5⋅ 5 2 (2+1) 3 + 2 3 + 1 3 6 2 − 15 2 − [2(5)] 2 5⋅ 5 2

Exercise 67

6 3 −2⋅ 10 2 2 2 + 18( 2 3 + 7 2 ) 2(19)− 3 3 6 3 −2⋅ 10 2 2 2 + 18( 2 3 + 7 2 ) 2(19)− 3 3

Solution

1070 11  or 97. 27 ¯ 1070 11  or 97. 27 ¯

Exercises for Review

Exercise 68

((Reference)) Use algebraic notation to write the statement "a number divided by eight, plus five, is equal to ten."

Exercise 69

((Reference)) Draw a number line that extends from −5 −5 to 5 and place points at all real numbers that are strictly greater than −3 −3 but less than or equal to 2.

Solution

A number line with arrows on each end, labeled from negative five to five in increments of one. There is a closed circle at two, and an open circle at negative three. These circles are connected by a black line.

Exercise 70

((Reference)) Is every integer a whole number?

Exercise 71

((Reference)) Use the commutative property of multiplication to write a number equal to the number yx yx .

Solution

xy xy

Exercise 72

((Reference)) Use the distributive property to expand 3(x+6) 3(x+6) .

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