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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

John left his house in Irvine at 8:35 am to drive to a meeting in Los Angeles, 45 miles away. He arrived at the meeting at 9:50. At 3:30 pm, he left the meeting and drove home. He arrived home at 5:18.
DaYoungan Reply
p-2/3=5/6 how do I solve it with explanation pls
Adedamola Reply
P=3/2
Vanarith
1/2p2-2/3p=5p/6
James
don't understand answer
Cindy
is y=7/5 a solution of 5y+3=10y-4
Adedamola Reply
yes
James
don't understand answer
Cindy
Lucinda has a pocketful of dimes and quarters with a value of $6.20. The number of dimes is 18 more than 3 times the number of quarters. How many dimes and how many quarters does Lucinda have?
Rhonda Reply
Find an equation for the line that passes through the point P ( 0 , − 4 ) and has a slope 8/9 .
Gabriel Reply
is that a negative 4 or positive 4?
Felix
y = mx + b
Felix
if negative -4, then -4=8/9(0) + b
Felix
-4=b
Felix
if positive 4, then 4=b
Felix
then plug in y=8/9x - 4 or y=8/9x+4
Felix
Macario is making 12 pounds of nut mixture with macadamia nuts and almonds. macadamia nuts cost $9 per pound and almonds cost $5.25 per pound. how many pounds of macadamia nuts and how many pounds of almonds should macario use for the mixture to cost $6.50 per pound to make?
Cherry Reply
Nga and Lauren bought a chest at a flea market for $50. They re-finished it and then added a 350 % mark - up
Makaila Reply
$1750
Cindy
the sum of two Numbers is 19 and their difference is 15
Abdulai Reply
2, 17
Jose
interesting
saw
4,2
Cindy
Felecia left her home to visit her daughter, driving 45mph. Her husband waited for the dog sitter to arrive and left home 20 minutes, or 13 hour later. He drove 55mph to catch up to Felecia. How long before he reaches her?
Rafi Reply
integer greater than 2 and less than 12
Emily Reply
2 < x < 12
Felix
I'm guessing you are doing inequalities...
Felix
Actually, translating words into algebraic expressions / equations...
Felix
hi
Darianna
hello
Mister
Eric here
Eric
6
Cindy
He charges $125 per job. His monthly expenses are $1,600. How many jobs must he work in order to make a profit of at least $2,400?
Alicia Reply
at least 20
Ayla
what are the steps?
Alicia
6.4 jobs
Grahame
32
Grahame
1600+2400= total amount with expenses. 4000/125= number of jobs needed to make that min profit of 2400. answer is 32
Orlando
He must work 32 jobs to make a profit
POP
32
Cindy
what is algebra
Azhar Reply
repeated addition and subtraction of the order of operations. i love algebra I'm obsessed.
Shemiah
hi
Krekar
Eric here. I'm a parent. 53 years old. I have never taken algebra. I want to learn.
Eric
I am 63 and never learned algebra
Cindy
One-fourth of the candies in a bag of M&M’s are red. If there are 23 red candies, how many candies are in the bag?
Leanna Reply
they are 92 candies in the bag
POP
92
Cindy
rectangular field solutions
Navin Reply
What is this?
Donna
t
muqtaar
the proudact of 3x^3-5×^2+3 and 2x^2+5x-4 in z7[x]/ is
anas Reply
?
Choli
a rock is thrown directly upward with an initial velocity of 96feet per second from a cliff 190 feet above a beach. The hight of tha rock above the beach after t second is given by the equation h=_16t^2+96t+190
Usman
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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