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1 x 3 y 2 z 1 . This fraction can be written without negative exponents by moving all the f a c t o r s 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

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Practice set b

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

x 3 y 4 z 8

x 3 y 4 z 8

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6 m 3 n 2 7 k 1

6 k 7 m 3 n 2

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1 a 2 b 6 c 8

a 2 b 6 c 8

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

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

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Sample set c

Rewrite 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

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Write 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

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Find the value of 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

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Practice set c

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

4 x 10 b 8

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

64 m 3 n 7 x 5

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

52

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Exercises

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

( x 5 ) 3

1 ( x 5 ) 3

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( a + 9 ) 10

1 ( a + 9 ) 10

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( a 1 ) 12

1 ( a 1 ) 12

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x 3 y 2 z 6

x 3 y 2 z 6

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a 7 b 9 z w 3

a 7 z w 3 b 9

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x 5 y 5 z 2

x 5 y 5 z 2

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x 4 y 8 z 3 w 4

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a 4 b 6 c 1 d 4

d 4 a 4 b 6 c

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x 9 y 6 z 1 w 5 r 2

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7 a 2 b 2 c 2

7 b 2 c 2 a 2

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

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

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

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

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

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

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2 ( a 8 ) 3 ( a 2 ) 5

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

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

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

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

1 ( 3 x 2 4 x 8 ) 9 ( 2 x + 11 ) 3

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

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

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

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( x 5 ) 4 3 b 2 c 4 ( x + 6 ) 8

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

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

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3 y 3 ( 9 x )

27 x y 3

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4 a 2 b 2 a 5 b 2

4 a 3

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

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2 3 x 2 2 3 x 2

1

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

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

1 ( x + 5 ) 4

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( a 4 ) 3 ( a 4 ) 10

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

8 ( b + 2 ) 9

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

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

8 a 5 b 2 c 2

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

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( 5 ) 2 ( 5 ) 1

5

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( 1 ) 1 ( 1 ) 1

1

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3 c 5 a 3 b 3

3 b 3 c 5 a 3

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

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

4 b c 9 y 2 a 2 d 3 z 8

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

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

4 a 3 b 2 c

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

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

3 x 3 y 2 z 4

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

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33 a 4 b 7 11 a 3 b 2

3 a 7 b 5

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51 x 5 y 3 3 x y

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

128 a 7 b x

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

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23 a 4 b 5 c 2 x 6 y 5

23 a 4 b 5 x 6 c 2 y 5

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

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10 x 3 y 7 3 x 5 z 2

10 3 x 2 y 7 z 2

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

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

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

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For the following problems, evaluate each numerical expression.

2 1 3 1 4 1

1 24

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10 2 + 3 ( 10 2 )

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2 1 + 4 1 2 2 + 4 2

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21 0 2 6 2 6 13

63

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

( m 0 ) 1 , m 0

1

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( x 6 y 6 z 1 ) 2

x 12 y 12 z 2

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( a 5 b 1 c 0 ) 6

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( y 3 x 4 ) 5

x 20 y 15

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( 2 a b 3 ) 4

16 a 4 b 12

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

a 6 x 4 25 b 12 y 18

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

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( r 5 s 4 m 8 n 7 ) 4

n 28 s 16 m 32 r 20

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( h 2 j 6 k 4 p ) 5

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Exercises for review

( [link] ) Simplify ( 4 x 5 y 3 z 0 ) 3

64 x 15 y 9

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( [link] ) Find the sum. 15 + 3 .

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( [link] ) Find the difference. 8 ( 12 ) .

20

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( [link] ) Simplify ( 3 ) ( 8 ) + 4 ( 5 ) .

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( [link] ) Find the value of m if m = 3 k 5 t k t + 6 when k = 4 and t = 2 .

1

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Source:  OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
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