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Basic Operations with Real Numbers: Negative Exponents

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

Summary: This module contains the basic operations with real numbers from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr.

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. This means that 4 and  1 4  are reciprocals. 4⋅ 1 4 =1. This means that 4 and  1 4  are reciprocals.

Example 2

6⋅ 1 6 =1. Hence, 6 and  1 6  are reciprocals. 6⋅ 1 6 =1. Hence, 6 and  1 6  are reciprocals.

Example 3

−2⋅ −1 2 =1. Hence, −2 and − 1 2  are reciprocals. −2⋅ −1 2 =1. Hence, −2 and − 1 2  are reciprocals.

Example 4

a⋅ 1 a =1. Hence, a and  1 a  are reciprocals if a≠0. a⋅ 1 a =1. Hence, a and  1 a  are reciprocals if a≠0.

Example 5

x⋅ 1 x =1. Hence, x and  1 x  are reciprocals if x≠0. x⋅ 1 x =1. Hence, x and  1 x  are reciprocals if x≠0.

Example 6

x 3 ⋅ 1 x 3 =1. Hence,  x 3  and  1 x 3  are reciprocals if x≠0. x 3 ⋅ 1 x 3 =1. Hence,  x 3  and  1 x 3  are reciprocals if x≠0.

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 x≠0 x≠0 .

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

(5x−1) −24 = 1 (5x−1) 24 (5x−1) −24 = 1 (5x−1) 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.

Exercise 1

y −5 y −5

Solution

1 y 5 1 y 5

Exercise 2

m −2 m −2

Solution

1 m 2 1 m 2

Exercise 3

3 −2 3 −2

Solution

1 9 1 9

Exercise 4

5 −1 5 −1

Solution

1 5 1 5

Exercise 5

2 −4 2 −4

Solution

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

(m−n) −(−4) (m−n) −(−4)

Solution

(m−n) 4 (m−n) 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 . The factor  x −2  can be moved from the numerator to the denominator by changing the exponent −2 to +2. x −2 y 5 = y 5 x 2 x −2 y 5 . The factor  x −2  can be moved from the numerator to the denominator by changing the exponent −2 to +2. x −2 y 5 = y 5 x 2
(1)

Example 14

a 9 b −3 . The factor  b −3  can be moved from the numerator to the denominator by changing the exponent −3 to +3. a 9 b −3 = a 9 b 3 a 9 b −3 . The factor  b −3  can be moved from the numerator to the denominator by changing the exponent −3 to +3. a 9 b −3 = a 9 b 3
(2)

Example 15

a 4 b 2 c −6 . This fraction can be written without any negative exponents by moving the factor  c −6  into the numerator. We must change the −6 to +6 to make the move legitimate. a 4 b 2 c −6 = a 4 b 2 c 6 a 4 b 2 c −6 . This fraction can be written without any negative exponents by moving the factor  c −6  into the numerator. We must change the −6 to +6 to make the move legitimate. a 4 b 2 c −6 = a 4 b 2 c 6
(3)

Example 16

1 x −3 y −2 z −1 . This fraction can be written without negative exponents by moving all the factors from the denominator to the numerator. Change the sign of each exponent: −3 to +3, −2 to +2, −1 to +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 . This fraction can be written without negative exponents by moving all the factors from the denominator to the numerator. Change the sign of each exponent: −3 to +3, −2 to +2, −1 to +1. 1 x −3 y −2 z −1 = x 3 y 2 z 1 = x 3 y 2 z
(4)

Practice Set B

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

Exercise 9

x −4 y 7 x −4 y 7

Solution

y 7 x 4 y 7 x 4

Exercise 10

a 2 b −4 a 2 b −4

Solution

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 (a−5b) −2 5b (a−4b) 5 3a (a−5b) −2 5b (a−4b) 5

Solution

3a 5b (a−5b) 2 (a−4b) 5 3a 5b (a−5b) 2 (a−4b) 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 7−4 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 7−4 b 9−(−6) = 3 a 3 b 9+6 = 3 a 3 b 15
(5)

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
(6)

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 = 1⋅100+3⋅64 = 100+192 = 292 1 10 −2 + 3 4 −3 = 1⋅ 10 2 +3⋅ 4 3 = 1⋅100+3⋅64 = 100+192 = 292
(7)

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 .

Solution

52

Exercises

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

Exercise 18

x −2 x −2

Solution

1x21x2

Exercise 19

x −4 x −4

Exercise 20

x −7 x −7

Solution

1x71x7

Exercise 21

a −8 a −8

Exercise 22

a −10 a −10

Solution

1a101a10

Exercise 23

b −12 b −12

Exercise 24

b −14 b −14

Solution

1b141b14

Exercise 25

y −1 y −1

Exercise 26

y −5 y −5

Solution

1y51y5

Exercise 27

(x+1) −2 (x+1) −2

Exercise 28

(x−5) −3 (x−5) −3

Solution

1(x−5)31(x−5)3

Exercise 29

(y−4) −6 (y−4) −6

Exercise 30

(a+9) −10 (a+9) −10

Solution

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

Exercise 31

(r+3) −8 (r+3) −8

Exercise 32

(a−1) −12 (a−1) −12

Solution

1(a−1)121(a−1)12

Exercise 33

x 3 y −2 x 3 y −2

Exercise 34

x 7 y −5 x 7 y −5

Solution

x7y5x7y5

Exercise 35

a 4 b −1 a 4 b −1

Exercise 36

a 7 b −8 a 7 b −8

Solution

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

Solution

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

Solution

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

Solution

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

Solution

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

Solution

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

Solution

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 (a−4) 3 b −6 c −7 7 a 2 (a−4) 3 b −6 c −7

Solution

7a2(a−4)3b6c77a2(a−4)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 (a−8) −3 (a−2) 5 2 (a−8) −3 (a−2) 5

Exercise 54

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

Solution

(x2+3)3(x2−1)4(x2+3)3(x2−1)4

Exercise 55

( x 4 +2x−1) −6 (x+5) 4 ( x 4 +2x−1) −6 (x+5) 4

Exercise 56

(3 x 2 −4x−8) −9 (2x+11) −3 (3 x 2 −4x−8) −9 (2x+11) −3

Solution

1(3x2−4x−8)9(2x+11)31(3x2−4x−8)9(2x+11)3

Exercise 57

(5 y 2 +8y−6) −2 (6y−1) −7 (5 y 2 +8y−6) −2 (6y−1) −7

Exercise 58

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

Solution

7a(a2−4)2(b2−1)27a(a2−4)2(b2−1)2

Exercise 59

(x−5) −4 3 b 2 c 4 (x+6) 8 (x−5) −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(y3−1)25y3(y3+1)z4w2(y3−1)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)

Solution

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

Solution

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

Solution

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

Solution

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

Exercise 69

(a−4) 3 (a−4) −10 (a−4) 3 (a−4) −10

Exercise 70

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

Solution

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 )

Solution

−8a5b2c2−8a5b2c2

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

Solution

−5−5

Exercise 75

(−9) −3 (9) 3 (−9) −3 (9) 3

Exercise 76

(−1) −1 (−1) −1 (−1) −1 (−1) −1

Solution

1

Exercise 77

(4) 2 (2) −4 (4) 2 (2) −4

Exercise 78

1 a −4 1 a −4

Solution

a4a4

Exercise 79

1 a −1 1 a −1

Exercise 80

4 x −6 4 x −6

Solution

4x64x6

Exercise 81

7 x −8 7 x −8

Exercise 82

23 y −1 23 y −1

Solution

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

Solution

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

Solution

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

Solution

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

Solution

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

Solution

128a7bx128a7bx

Exercise 95

(x+3) 3 (y−6) 4 (x+3) 5 (y−6) −8 (x+3) 3 (y−6) 4 (x+3) 5 (y−6) −8

Exercise 96

4 x 3 y 7 4 x 3 y 7

Solution

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 (x−5) 9 −1 (x+5) 3 3 x 2 y −2 (x−5) 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.

Exercise 103

4 −1 4 −1

Exercise 104

7 −1 7 −1

Solution

1717

Exercise 105

6 −2 6 −2

Exercise 106

2 −5 2 −5

Solution

132132

Exercise 107

3 −4 3 −4

Exercise 108

6⋅ 3 −3 6⋅ 3 −3

Solution

2929

Exercise 109

4⋅ 9 −2 4⋅ 9 −2

Exercise 110

28⋅ 14 −1 28⋅ 14 −1

Solution

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

Solution

124124

Exercise 113

10 −2 +3( 10 −2 ) 10 −2 +3( 10 −2 )

Exercise 114

(−3) −2 (−3) −2

Solution

1919

Exercise 115

(−10) −1 (−10) −1

Exercise 116

3 2 −3 3 2 −3

Solution

24

Exercise 117

4 −1 5 −2 4 −1 5 −2

Exercise 118

2 4 −7 4 −1 2 4 −7 4 −1

Solution

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 2⋅6−13 21 0 − 2 6 2⋅6−13

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

Solution

1a151a15

Exercise 123

( x 7 ) −4 ( x 7 ) −4

Exercise 124

( x 4 ) −8 ( x 4 ) −8

Solution

1x321x32

Exercise 125

( b −2 ) 7 ( b −2 ) 7

Exercise 126

( b −4 ) −1 ( b −4 ) −1

Solution

b4b4

Exercise 127

( y −3 ) −4 ( y −3 ) −4

Exercise 128

( y −9 ) −3 ( y −9 ) −3

Solution

y27y27

Exercise 129

( a −1 ) −1 ( a −1 ) −1

Exercise 130

( b −1 ) −1 ( b −1 ) −1

Solution

bb

Exercise 131

( a 0 ) −1 , a≠0 ( a 0 ) −1 , a≠0

Exercise 132

( m 0 ) −1 , m≠0 ( m 0 ) −1 , m≠0

Solution

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

Solution

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

Solution

x20y15x20y15

Exercise 137

( a −8 b −6 ) 3 ( a −8 b −6 ) 3

Exercise 138

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

Solution

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

Solution

64x15y964x15y9

Exercise 145

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

Exercise 146

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

Solution

20

Exercise 147

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

Exercise 148

((Reference)) Find the value of m m if m= −3k−5t kt+6 m= −3k−5t kt+6 when k=4 k=4 and t=−2 t=−2 .

Solution

1

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