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# Basic Operations with Real Numbers: Negative 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 basic operations with real numbers are presented in this chapter. The concept of absolute value is discussed both geometrically and symbolically. The geometric presentation offers a visual understanding of the meaning of |x|. The symbolic presentation includes a literal explanation of how to use the definition. Negative exponents are developed, using reciprocals and the rules of exponents the student has already learned. Scientific notation is also included, using unique and real-life examples.

Objectives of this module: understand the concepts of reciprocals and negative exponents, be able to work with negative exponents.

Note: You are viewing an old version of this document. The latest version is available here.

## Overview

• Reciprocals
• Negative Exponents
• Working with Negative Exponents

## Reciprocals

### Reciprocals

Two real numbers are said to be reciprocals of each other if their product is 1. Every nonzero real number has exactly one reciprocal, as shown in the examples below. Zero has no reciprocal.

### Example 1

4 1 4 =1. Thismeansthat4and 1 4 arereciprocals. 4 1 4 =1. Thismeansthat4and 1 4 arereciprocals.

### Example 2

6 1 6 =1. Hence,6and 1 6 arereciprocals. 6 1 6 =1. Hence,6and 1 6 arereciprocals.

### Example 3

2 1 2 =1. Hence,2and 1 2 arereciprocals. 2 1 2 =1. Hence,2and 1 2 arereciprocals.

### Example 4

a 1 a =1. Hence,aand 1 a arereciprocalsifa0. a 1 a =1. Hence,aand 1 a arereciprocalsifa0.

### Example 5

x 1 x =1. Hence,xand 1 x arereciprocalsifx0. x 1 x =1. Hence,xand 1 x arereciprocalsifx0.

### Example 6

x 3 1 x 3 =1. Hence, x 3 and 1 x 3 arereciprocalsifx0. x 3 1 x 3 =1. Hence, x 3 and 1 x 3 arereciprocalsifx0.

## Negative Exponents

We can use the idea of reciprocals to find a meaning for negative exponents.

Consider the product of x 3 x 3 and x 3 x 3 . Assume x0 x0 .

x 3 x 3 = x 3+(3) = x 0 =1 x 3 x 3 = x 3+(3) = x 0 =1

Thus, since the product of x 3 x 3 and x 3 x 3 is 1, x 3 x 3 and x 3 x 3 must be reciprocals.

We also know that x 3 1 x 3 =1 x 3 1 x 3 =1 . (See problem 6 above.) Thus, x 3 x 3 and 1 x 3 1 x 3 are also reciprocals.

Then, since x 3 x 3 and 1 x 3 1 x 3 are both reciprocals of x 3 x 3 and a real number can have only one reciprocal, it must be that x 3 = 1 x 3 x 3 = 1 x 3 .

We have used 3 3 as the exponent, but the process works as well for all other negative integers. We make the following definition.

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

x n = 1 x n x n = 1 x n

## Sample Set A

Write each of the following so that only positive exponents appear.

### Example 7

x 6 = 1 x 6 x 6 = 1 x 6

### Example 8

a 1 = 1 a 1 = 1 a a 1 = 1 a 1 = 1 a

### Example 9

7 2 = 1 7 2 = 1 49 7 2 = 1 7 2 = 1 49

### Example 10

(3a) 6 = 1 (3a) 6 (3a) 6 = 1 (3a) 6

### Example 11

(5x1) 24 = 1 (5x1) 24 (5x1) 24 = 1 (5x1) 24

### Example 12

(k+2z) (8) = (k+2z) 8 (k+2z) (8) = (k+2z) 8

## Practice Set A

Write each of the following using only positive exponents.

y 5 y 5

1 y 5 1 y 5

m 2 m 2

1 m 2 1 m 2

3 2 3 2

1 9 1 9

5 1 5 1

1 5 1 5

2 4 2 4

1 16 1 16

### Exercise 6

(xy) 4 (xy) 4

#### Solution

1 (xy) 4 1 (xy) 4

### Exercise 7

(a+2b) 12 (a+2b) 12

#### Solution

1 (a+2b) 12 1 (a+2b) 12

### Exercise 8

(mn) (4) (mn) (4)

(mn) 4 (mn) 4

### CAUTION

It is important to note that a n a n is not necessarily a negative number. For example,

3 2 = 1 3 2 = 1 9 3 2 9 3 2 = 1 3 2 = 1 9 3 2 9

## Working with Negative Exponents

The problems of Sample Set A suggest the following rule for working with exponents:

### Moving Factors Up and Down

In a fraction, a factor can be moved from the numerator to the denominator or from the denominator to the numerator by changing the sign of the exponent.

## Sample Set B

Write each of the following so that only positive exponents appear.

### Example 13

x 2 y 5 . Thefactor x 2 canbemovedfromthenumeratortothe denominatorbychangingtheexponent2to+2. x 2 y 5 = y 5 x 2 x 2 y 5 . Thefactor x 2 canbemovedfromthenumeratortothe denominatorbychangingtheexponent2to+2. x 2 y 5 = y 5 x 2

### Example 14

a 9 b 3 . Thefactor b 3 canbemovedfromthenumeratortothe denominatorbychangingtheexponent3to+3. a 9 b 3 = a 9 b 3 a 9 b 3 . Thefactor b 3 canbemovedfromthenumeratortothe denominatorbychangingtheexponent3to+3. a 9 b 3 = a 9 b 3

### Example 15

a 4 b 2 c 6 . Thisfractioncanbewrittenwithoutanynegativeexponents bymovingthefactor c 6 intothenumerator. Wemustchangethe6to+6tomakethemovelegitimate. a 4 b 2 c 6 = a 4 b 2 c 6 a 4 b 2 c 6 . Thisfractioncanbewrittenwithoutanynegativeexponents bymovingthefactor c 6 intothenumerator. Wemustchangethe6to+6tomakethemovelegitimate. a 4 b 2 c 6 = a 4 b 2 c 6

### Example 16

1 x 3 y 2 z 1 . Thisfractioncanbewrittenwithoutnegativeexponents bymovingallthefactorsfromthedenominatorto thenumerator.Changethesignofeachexponent:3to+3, 2to+2,1to+1. 1 x 3 y 2 z 1 = x 3 y 2 z 1 = x 3 y 2 z 1 x 3 y 2 z 1 . Thisfractioncanbewrittenwithoutnegativeexponents bymovingallthefactorsfromthedenominatorto thenumerator.Changethesignofeachexponent:3to+3, 2to+2,1to+1. 1 x 3 y 2 z 1 = x 3 y 2 z 1 = x 3 y 2 z

## Practice Set B

Write each of the following so that only positive exponents appear.

x 4 y 7 x 4 y 7

y 7 x 4 y 7 x 4

a 2 b 4 a 2 b 4

a 2 b 4 a 2 b 4

### Exercise 11

x 3 y 4 z 8 x 3 y 4 z 8

#### Solution

x 3 y 4 z 8 x 3 y 4 z 8

### Exercise 12

6 m 3 n 2 7 k 1 6 m 3 n 2 7 k 1

#### Solution

6k 7 m 3 n 2 6k 7 m 3 n 2

### Exercise 13

1 a 2 b 6 c 8 1 a 2 b 6 c 8

#### Solution

a 2 b 6 c 8 a 2 b 6 c 8

### Exercise 14

3a (a5b) 2 5b (a4b) 5 3a (a5b) 2 5b (a4b) 5

#### Solution

3a 5b (a5b) 2 (a4b) 5 3a 5b (a5b) 2 (a4b) 5

## Sample Set C

### Example 17

Rewrite 24 a 7 b 9 2 3 a 4 b 6 24 a 7 b 9 2 3 a 4 b 6 in a simpler form.

Notice that we are dividing powers with the same base. We’ll proceed by using the rules of exponents.

24 a 7 b 9 2 3 a 4 b 6 = 24 a 7 b 9 8 a 4 b 6 = 3 a 74 b 9(6) = 3 a 3 b 9+6 = 3 a 3 b 15 24 a 7 b 9 2 3 a 4 b 6 = 24 a 7 b 9 8 a 4 b 6 = 3 a 74 b 9(6) = 3 a 3 b 9+6 = 3 a 3 b 15

### Example 18

Write 9 a 5 b 3 5 x 3 y 2 9 a 5 b 3 5 x 3 y 2 so that no denominator appears.

We can eliminate the denominator by moving all factors that make up the denominator to the numerator.

9 a 5 b 3 5 1 x 3 y 2 9 a 5 b 3 5 1 x 3 y 2

### Example 19

Find the value of 1 10 2 + 3 4 3 1 10 2 + 3 4 3 .

We can evaluate this expression by eliminating the negative exponents.

1 10 2 + 3 4 3 = 1 10 2 +3 4 3 = 1100+364 = 100+192 = 292 1 10 2 + 3 4 3 = 1 10 2 +3 4 3 = 1100+364 = 100+192 = 292

## Practice Set C

### Exercise 15

Rewrite 36 x 8 b 3 3 2 x 2 b 5 36 x 8 b 3 3 2 x 2 b 5 in a simpler form.

#### Solution

4 x 10 b 8 4 x 10 b 8

### Exercise 16

Write 2 4 m 3 n 7 4 1 x 5 2 4 m 3 n 7 4 1 x 5 in a simpler form and one in which no denominator appears.

#### Solution

64 m 3 n 7 x 5 64 m 3 n 7 x 5

### Exercise 17

Find the value of 2 5 2 + 6 2 2 3 3 2 2 5 2 + 6 2 2 3 3 2 .

52

## Exercises

Write the following expressions using only positive exponents. Assume all variables are nonzero.

x 2 x 2

1x21x2

x 4 x 4

x 7 x 7

1x71x7

a 8 a 8

a 10 a 10

1a101a10

b 12 b 12

b 14 b 14

1b141b14

y 1 y 1

y 5 y 5

1y51y5

(x+1) 2 (x+1) 2

(x5) 3 (x5) 3

1(x5)31(x5)3

(y4) 6 (y4) 6

### Exercise 30

(a+9) 10 (a+9) 10

1(a+9)101(a+9)10

(r+3) 8 (r+3) 8

(a1) 12 (a1) 12

1(a1)121(a1)12

x 3 y 2 x 3 y 2

x 7 y 5 x 7 y 5

x7y5x7y5

a 4 b 1 a 4 b 1

a 7 b 8 a 7 b 8

a7b8a7b8

### Exercise 37

a 2 b 3 c 2 a 2 b 3 c 2

### Exercise 38

x 3 y 2 z 6 x 3 y 2 z 6

x3y2z6x3y2z6

### Exercise 39

x 3 y 4 z 2 w x 3 y 4 z 2 w

### Exercise 40

a 7 b 9 z w 3 a 7 b 9 z w 3

a7zw3b9a7zw3b9

### Exercise 41

a 3 b 1 z w 2 a 3 b 1 z w 2

### Exercise 42

x 5 y 5 z 2 x 5 y 5 z 2

x5y5z2x5y5z2

### Exercise 43

x 4 y 8 z 3 w 4 x 4 y 8 z 3 w 4

### Exercise 44

a 4 b 6 c 1 d 4 a 4 b 6 c 1 d 4

d4a4b6cd4a4b6c

### Exercise 45

x 9 y 6 z 1 w 5 r 2 x 9 y 6 z 1 w 5 r 2

### Exercise 46

4 x 6 y 2 4 x 6 y 2

4y2x64y2x6

### Exercise 47

5 x 2 y 2 z 5 5 x 2 y 2 z 5

### Exercise 48

7 a 2 b 2 c 2 7 a 2 b 2 c 2

7b2c2a27b2c2a2

### Exercise 49

4 x 3 (x+1) 2 y 4 z 1 4 x 3 (x+1) 2 y 4 z 1

### Exercise 50

7 a 2 (a4) 3 b 6 c 7 7 a 2 (a4) 3 b 6 c 7

#### Solution

7a2(a4)3b6c77a2(a4)3b6c7

### Exercise 51

18 b 6 ( b 2 3) 5 c 4 d 5 e 1 18 b 6 ( b 2 3) 5 c 4 d 5 e 1

### Exercise 52

7 (w+2) 2 (w+1) 3 7 (w+2) 2 (w+1) 3

#### Solution

7(w+1)3(w+2)27(w+1)3(w+2)2

### Exercise 53

2 (a8) 3 (a2) 5 2 (a8) 3 (a2) 5

### Exercise 54

( x 2 +3) 3 ( x 2 1) 4 ( x 2 +3) 3 ( x 2 1) 4

#### Solution

(x2+3)3(x21)4(x2+3)3(x21)4

### Exercise 55

( x 4 +2x1) 6 (x+5) 4 ( x 4 +2x1) 6 (x+5) 4

### Exercise 56

(3 x 2 4x8) 9 (2x+11) 3 (3 x 2 4x8) 9 (2x+11) 3

#### Solution

1(3x24x8)9(2x+11)31(3x24x8)9(2x+11)3

### Exercise 57

(5 y 2 +8y6) 2 (6y1) 7 (5 y 2 +8y6) 2 (6y1) 7

### Exercise 58

7a ( a 2 4) 2 ( b 2 1) 2 7a ( a 2 4) 2 ( b 2 1) 2

#### Solution

7a(a24)2(b21)27a(a24)2(b21)2

### Exercise 59

(x5) 4 3 b 2 c 4 (x+6) 8 (x5) 4 3 b 2 c 4 (x+6) 8

### Exercise 60

( y 3 +1) 1 5 y 3 z 4 w 2 ( y 3 1) 2 ( y 3 +1) 1 5 y 3 z 4 w 2 ( y 3 1) 2

#### Solution

5y3(y3+1)z4w2(y31)25y3(y3+1)z4w2(y31)2

### Exercise 61

5 x 3 (2 x 7 ) 5 x 3 (2 x 7 )

### Exercise 62

3 y 3 (9x) 3 y 3 (9x)

27xy327xy3

### Exercise 63

6 a 4 (2 a 6 ) 6 a 4 (2 a 6 )

### Exercise 64

4 a 2 b 2 a 5 b 2 4 a 2 b 2 a 5 b 2

4a34a3

### Exercise 65

5 1 a 2 b 6 b 11 c 3 c 9 5 1 a 2 b 6 b 11 c 3 c 9

### Exercise 66

2 3 x 2 2 3 x 2 2 3 x 2 2 3 x 2

1

### Exercise 67

7 a 3 b 9 5a b 6 c 2 c 4 7 a 3 b 9 5a b 6 c 2 c 4

### Exercise 68

(x+5) 2 (x+5) 6 (x+5) 2 (x+5) 6

1(x+5)41(x+5)4

### Exercise 69

(a4) 3 (a4) 10 (a4) 3 (a4) 10

### Exercise 70

8 (b+2) 8 (b+2) 4 (b+2) 3 8 (b+2) 8 (b+2) 4 (b+2) 3

8(b+2)98(b+2)9

### Exercise 71

3 a 5 b 7 ( a 2 +4) 3 6 a 4 b ( a 2 +4) 1 ( a 2 +4) 3 a 5 b 7 ( a 2 +4) 3 6 a 4 b ( a 2 +4) 1 ( a 2 +4)

### Exercise 72

4 a 3 b 5 (2 a 2 b 7 c 2 ) 4 a 3 b 5 (2 a 2 b 7 c 2 )

8a5b2c28a5b2c2

### Exercise 73

2 x 2 y 4 z 4 (6 x 3 y 3 z) 2 x 2 y 4 z 4 (6 x 3 y 3 z)

### Exercise 74

(5) 2 (5) 1 (5) 2 (5) 1

55

### Exercise 75

(9) 3 (9) 3 (9) 3 (9) 3

### Exercise 76

(1) 1 (1) 1 (1) 1 (1) 1

1

### Exercise 77

(4) 2 (2) 4 (4) 2 (2) 4

1 a 4 1 a 4

a4a4

1 a 1 1 a 1

4 x 6 4 x 6

4x64x6

7 x 8 7 x 8

23 y 1 23 y 1

23y23y

### Exercise 83

6 a 2 b 4 6 a 2 b 4

### Exercise 84

3 c 5 a 3 b 3 3 c 5 a 3 b 3

3b3c5a33b3c5a3

### Exercise 85

16 a 2 b 6 c 2y z 5 w 4 16 a 2 b 6 c 2y z 5 w 4

### Exercise 86

24 y 2 z 8 6 a 2 b 1 c 9 d 3 24 y 2 z 8 6 a 2 b 1 c 9 d 3

#### Solution

4bc9y2a2d3z84bc9y2a2d3z8

### Exercise 87

3 1 b 5 (b+7) 4 9 1 a 4 (a+7) 2 3 1 b 5 (b+7) 4 9 1 a 4 (a+7) 2

### Exercise 88

36 a 6 b 5 c 8 3 2 a 3 b 7 c 9 36 a 6 b 5 c 8 3 2 a 3 b 7 c 9

4a3b2c4a3b2c

### Exercise 89

45 a 4 b 2 c 6 15 a 2 b 7 c 8 45 a 4 b 2 c 6 15 a 2 b 7 c 8

### Exercise 90

3 3 x 4 y 3 z 3 2 x y 5 z 5 3 3 x 4 y 3 z 3 2 x y 5 z 5

3x3y2z43x3y2z4

### Exercise 91

21 x 2 y 2 z 5 w 4 7xy z 12 w 14 21 x 2 y 2 z 5 w 4 7xy z 12 w 14

### Exercise 92

33 a 4 b 7 11 a 3 b 2 33 a 4 b 7 11 a 3 b 2

3a7b53a7b5

### Exercise 93

51 x 5 y 3 3xy 51 x 5 y 3 3xy

### Exercise 94

2 6 x 5 y 2 a 7 b 5 2 1 x 4 y 2 b 6 2 6 x 5 y 2 a 7 b 5 2 1 x 4 y 2 b 6

128a7bx128a7bx

### Exercise 95

(x+3) 3 (y6) 4 (x+3) 5 (y6) 8 (x+3) 3 (y6) 4 (x+3) 5 (y6) 8

### Exercise 96

4 x 3 y 7 4 x 3 y 7

4x3y74x3y7

### Exercise 97

5 x 4 y 3 a 3 5 x 4 y 3 a 3

### Exercise 98

23 a 4 b 5 c 2 x 6 y 5 23 a 4 b 5 c 2 x 6 y 5

#### Solution

23a4b5x6c2y523a4b5x6c2y5

### Exercise 99

2 3 b 5 c 2 d 9 4 b 4 cx 2 3 b 5 c 2 d 9 4 b 4 cx

### Exercise 100

10 x 3 y 7 3 x 5 z 2 10 x 3 y 7 3 x 5 z 2

#### Solution

103x2y7z2103x2y7z2

### Exercise 101

3 x 2 y 2 (x5) 9 1 (x+5) 3 3 x 2 y 2 (x5) 9 1 (x+5) 3

### Exercise 102

14 a 2 b 2 c 12 ( a 2 +21) 4 4 2 a 2 b 1 (a+6) 3 14 a 2 b 2 c 12 ( a 2 +21) 4 4 2 a 2 b 1 (a+6) 3

#### Solution

224b3c12(a2+21)4(a+6)3224b3c12(a2+21)4(a+6)3

For the following problems, evaluate each numerical expression.

4 1 4 1

7 1 7 1

1717

6 2 6 2

2 5 2 5

132132

3 4 3 4

6 3 3 6 3 3

2929

4 9 2 4 9 2

28 14 1 28 14 1

2

### Exercise 111

2 3 ( 3 2 ) 2 3 ( 3 2 )

### Exercise 112

2 1 3 1 4 1 2 1 3 1 4 1

124124

### Exercise 113

10 2 +3( 10 2 ) 10 2 +3( 10 2 )

(3) 2 (3) 2

1919

(10) 1 (10) 1

3 2 3 3 2 3

24

4 1 5 2 4 1 5 2

### Exercise 118

2 4 7 4 1 2 4 7 4 1

36

### Exercise 119

2 1 + 4 1 2 2 + 4 2 2 1 + 4 1 2 2 + 4 2

### Exercise 120

21 0 2 6 2613 21 0 2 6 2613

#### Solution

63

For the following problems, write each expression so that only positive exponents appear.

### Exercise 121

( a 6 ) 2 ( a 6 ) 2

### Exercise 122

( a 5 ) 3 ( a 5 ) 3

1a151a15

### Exercise 123

( x 7 ) 4 ( x 7 ) 4

### Exercise 124

( x 4 ) 8 ( x 4 ) 8

1x321x32

### Exercise 125

( b 2 ) 7 ( b 2 ) 7

### Exercise 126

( b 4 ) 1 ( b 4 ) 1

b4b4

### Exercise 127

( y 3 ) 4 ( y 3 ) 4

### Exercise 128

( y 9 ) 3 ( y 9 ) 3

y27y27

### Exercise 129

( a 1 ) 1 ( a 1 ) 1

### Exercise 130

( b 1 ) 1 ( b 1 ) 1

bb

### Exercise 131

( a 0 ) 1 , a0 ( a 0 ) 1 , a0

### Exercise 132

( m 0 ) 1 , m0 ( m 0 ) 1 , m0

1

### Exercise 133

( x 3 y 7 ) 4 ( x 3 y 7 ) 4

### Exercise 134

( x 6 y 6 z 1 ) 2 ( x 6 y 6 z 1 ) 2

x12y12z2x12y12z2

### Exercise 135

( a 5 b 1 c 0 ) 6 ( a 5 b 1 c 0 ) 6

### Exercise 136

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

x20y15x20y15

### Exercise 137

( a 8 b 6 ) 3 ( a 8 b 6 ) 3

### Exercise 138

( 2a b 3 ) 4 ( 2a b 3 ) 4

16a4b1216a4b12

### Exercise 139

( 3b a 2 ) 5 ( 3b a 2 ) 5

### Exercise 140

( 5 1 a 3 b 6 x 2 y 9 ) 2 ( 5 1 a 3 b 6 x 2 y 9 ) 2

#### Solution

a6x425b12y18a6x425b12y18

### Exercise 141

( 4 m 3 n 6 2 m 5 n ) 3 ( 4 m 3 n 6 2 m 5 n ) 3

### Exercise 142

( r 5 s 4 m 8 n 7 ) 4 ( r 5 s 4 m 8 n 7 ) 4

#### Solution

n28s16m32r20n28s16m32r20

### Exercise 143

( h 2 j 6 k 4 p ) 5 ( h 2 j 6 k 4 p ) 5

## Exercises for Review

### Exercise 144

((Reference)) Simplify (4 x 5 y 3 z 0 ) 3 (4 x 5 y 3 z 0 ) 3

64x15y964x15y9

### Exercise 145

((Reference)) Find the sum. 15+3 15+3 .

### Exercise 146

((Reference)) Find the difference. 8(12) 8(12) .

20

### Exercise 147

((Reference)) Simplify (3)(8)+4(5) (3)(8)+4(5) .

### Exercise 148

((Reference)) Find the value of m m if m= 3k5t kt+6 m= 3k5t kt+6 when k=4 k=4 and t=2 t=2 .

1

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