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To get from the original fraction raised to a negative exponent to the final result, we took the reciprocal of the base—the fraction—and changed the sign of the exponent.

This leads us to the Quotient to a Negative Power Property .

Quotient to a negative exponent property

If a and b are real numbers, a 0 , b 0 , and n is an integer, then ( a b ) n = ( b a ) n .

Simplify: ( 5 7 ) −2 ( 2 x y ) −3 .

Solution


  1. ( 5 7 ) −2 Use the Quotient to a Negative Exponent Property, ( a b ) n = ( b a ) n . Take the reciprocal of the fraction and change the sign of the exponent. ( 7 5 ) 2 Simplify. 49 25


  2. ( 2 x y ) −3 Use the Quotient to a Negative Exponent Property, ( a b ) n = ( b a ) n . Take the reciprocal of the fraction and change the sign of the exponent. ( y 2 x ) 3 Simplify. y 3 8 x 3
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Simplify: ( 2 3 ) −4 ( 6 m n ) −2 .

81 16 n 2 36 m 2

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Simplify: ( 3 5 ) −3 ( a 2 b ) −4 .

125 27 16 b 4 a 4

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When simplifying an expression with exponents, we must be careful to correctly identify the base.

Simplify: ( −3 ) −2 3 −2 ( 1 3 ) −2 ( 1 3 ) −2 .

Solution

  1. Here the exponent applies to the base −3 .
    ( −3 ) −2 Take the reciprocal of the base and change the sign of the exponent. 1 ( −3 ) −2 Simplify. 1 9

  2. The expression 3 −2 means “find the opposite of 3 −2 ”. Here the exponent applies to the base 3.
    3 −2 Rewrite as a product with −1 . −1 · 3 −2 Take the reciprocal of the base and change the sign of the exponent. −1 · 1 3 2 Simplify. 1 9

  3. Here the exponent applies to the base ( 1 3 ) .
    ( 1 3 ) −2 Take the reciprocal of the base and change the sign of the exponent. ( 3 1 ) 2 Simplify. 9

  4. The expression ( 1 3 ) −2 means “find the opposite of ( 1 3 ) −2 ”. Here the exponent applies to the base ( 1 3 ) .
    ( 1 3 ) −2 Rewrite as a product with −1 . −1 · ( 1 3 ) −2 Take the reciprocal of the base and change the sign of the exponent. −1 · ( 3 1 ) 2 Simplify. −9
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Simplify: ( −5 ) −2 5 −2 ( 1 5 ) −2 ( 1 5 ) −2 .

1 25 1 25 25 −25

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Simplify: ( −7 ) −2 7 −2 , ( 1 7 ) −2 ( 1 7 ) −2 .

1 49 1 49 49 −49

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We must be careful to follow the Order of Operations. In the next example, parts (a) and (b) look similar, but the results are different.

Simplify: 4 · 2 −1 ( 4 · 2 ) −1 .

Solution


  1. Do exponents before multiplication. 4 · 2 −1 Use a n = 1 a n . 4 · 1 2 1 Simplify. 2


  2. ( 4 · 2 ) −1 Simplify inside the parentheses first. ( 8 ) −1 Use a n = 1 a n . 1 8 1 Simplify. 1 8
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Simplify: 6 · 3 −1 ( 6 · 3 ) −1 .

2 1 18

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Simplify: 8 · 2 −2 ( 8 · 2 ) −2 .

2 1 16

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When a variable is raised to a negative exponent, we apply the definition the same way we did with numbers. We will assume all variables are non-zero.

Simplify: x −6 ( u 4 ) −3 .

Solution


  1. x −6 Use the definition of a negative exponent, a n = 1 a n . 1 x 6


  2. ( u 4 ) −3 Use the definition of a negative exponent, a n = 1 a n . 1 ( u 4 ) 3 Simplify. 1 u 12
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Simplify: y −7 ( z 3 ) −5 .

1 y 7 1 z 15

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Simplify: p −9 ( q 4 ) −6 .

1 p 9 1 q 24

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When there is a product and an exponent we have to be careful to apply the exponent to the correct quantity. According to the Order of Operations, we simplify expressions in parentheses before applying exponents. We’ll see how this works in the next example.

Simplify: 5 y −1 ( 5 y ) −1 ( −5 y ) −1 .

Solution


  1. 5 y −1 Notice the exponent applies to just the base y . Take the reciprocal of y and change the sign of the exponent. 5 · 1 y 1 Simplify. 5 y


  2. ( 5 y ) −1 Here the parentheses make the exponent apply to the base 5 y . Take the reciprocal of 5 y and change the sign of the exponent. 1 ( 5 y ) 1 Simplify. 1 5 y


  3. ( −5 y ) −1 The base here is −5 y . Take the reciprocal of −5 y and change the sign of the exponent. 1 ( −5 y ) 1 Simplify. 1 −5 y Use a b = a b . 1 5 y
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Questions & Answers

divide 3x⁴-4x³-3x-1 by x-3
Ritik Reply
how to multiply the monomial
Ceny Reply
Two sisters like to compete on their bike rides. Tamara can go 4 mph faster than her sister, Samantha. If it takes Samantha 1 hours longer than Tamara to go 80 miles, how fast can Samantha ride her bike? Got questions? Get instant answers now!
Seera Reply
how do u solve that question
Seera
Two sisters like to compete on their bike rides. Tamara can go 4 mph faster than her sister, Samantha. If it takes Samantha 1 hours longer than Tamara to go 80 miles, how fast can Samantha ride her bike?
Seera
Speed=distance ÷ time
Tremayne
x-3y =1; 3x-2y+4=0 graph
Juned Reply
Brandon has a cup of quarters and dimes with a total of 5.55$. The number of quarters is five less than three times the number of dimes
ashley Reply
app is wrong how can 350 be divisible by 3.
Raheem Reply
June needs 48 gallons of punch for a party and has two different coolers to carry it in. The bigger cooler is five times as large as the smaller cooler. How many gallons can each cooler hold?
Susanna Reply
Susanna if the first cooler holds five times the gallons then the other cooler. The big cooler holda 40 gallons and the 2nd will hold 8 gallons is that correct?
Georgie
@Susanna that person is correct if you divide 40 by 8 you can see it's 5 it's simple
Ashley
@Geogie my bad that was meant for u
Ashley
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Lorris Reply
I'm getting "math processing error" on math problems. Anyone know why?
Ray Reply
Can you all help me I don't get any of this
Jade Reply
4^×=9
Alberto Reply
Did anyone else have trouble getting in quiz link for linear inequalities?
Sireka Reply
operation of trinomial
Justin Reply
y=2×+9
Jacob Reply
Keshad gets paid $2,400 per month plus 6% of his sales. His brother earns $3,300 per month. For what amount of total sales will Keshad’s monthly pay be higher than his brother’s monthly pay?
Hector Reply
Practice Key Terms 2

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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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