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Simplify: 8 p −1 ( 8 p ) −1 ( −8 p ) −1 .

8 p 1 8 p 1 8 p

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Simplify: 11 q −1 ( 11 q ) −1 ( 11 q ) −1 ( −11 q ) −1 .

1 11 q 1 11 q 1 11 q 1 11 q

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With negative exponents, the Quotient Rule needs only one form a m a n = a m n , for a 0 . When the exponent in the denominator is larger than the exponent in the numerator, the exponent of the quotient will be negative.

Simplify expressions with integer exponents

All of the exponent properties we developed earlier in the chapter with whole number exponents apply to integer exponents, too. We restate them here for reference.

Summary of exponent properties

If a and b are real numbers, and m and n are integers, then

Product Property a m · a n = a m + n Power Property ( a m ) n = a m · n Product to a Power ( a b ) m = a m b m Quotient Property a m a n = a m n , a 0 Zero Exponent Property a 0 = 1 , a 0 Quotient to a Power Property ( a b ) m = a m b m , b 0 Properties of Negative Exponents a n = 1 a n and 1 a n = a n Quotient to a Negative Exponent ( a b ) n = ( b a ) n

Simplify: x −4 · x 6 y −6 · y 4 z −5 · z −3 .

Solution


  1. x −4 · x 6 Use the Product Property, a m · a n = a m + n . x −4 + 6 Simplify. x 2


  2. y −6 · y 4 Notice the same bases, so add the exponents. y −6 + 4 Simplify. y −2 Use the definition of a negative exponent, a n = 1 a n . 1 y 2


  3. z −5 · z −3 Add the exponents, since the bases are the same. z −5 3 Simplify. z −8 Take the reciprocal and change the sign of the exponent, using the definition of a negative exponent. 1 z 8
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Simplify: x −3 · x 7 y −7 · y 2 z −4 · z −5 .

x 4 1 y 5 1 z 9

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Simplify: a −1 · a 6 b −8 · b 4 c −8 · c −7 .

a 5 1 b 4 1 c 15

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In the next two examples, we’ll start by using the Commutative Property to group the same variables together. This makes it easier to identify the like bases before using the Product Property.

Simplify: ( m 4 n −3 ) ( m −5 n −2 ) .

Solution

( m 4 n −3 ) ( m −5 n −2 ) Use the Commutative Property to get like bases together. m 4 m −5 · n −2 n −3 Add the exponents for each base. m −1 · n −5 Take reciprocals and change the signs of the exponents. 1 m 1 · 1 n 5 Simplify. 1 m n 5

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Simplify: ( p 6 q −2 ) ( p −9 q −1 ) .

1 p 3 q 3

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Simplify: ( r 5 s −3 ) ( r −7 s −5 ) .

1 r 2 s 8

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If the monomials have numerical coefficients, we multiply the coefficients, just like we did earlier.

Simplify: ( 2 x −6 y 8 ) ( −5 x 5 y −3 ) .

Solution

( 2 x −6 y 8 ) ( −5 x 5 y −3 ) Rewrite with the like bases together. 2 ( −5 ) · ( x −6 x 5 ) · ( y 8 y −3 ) Multiply the coefficients and add the exponents of each variable. −10 · x −1 · y 5 Use the definition of a negative exponent, a n = 1 a n . −10 · 1 x 1 · y 5 Simplify. −10 y 5 x

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Simplify: ( 3 u −5 v 7 ) ( −4 u 4 v −2 ) .

12 v 5 u

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Simplify: ( −6 c −6 d 4 ) ( −5 c −2 d −1 ) .

30 d 3 c 8

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In the next two examples, we’ll use the Power Property and the Product to a Power Property.

Simplify: ( 6 k 3 ) −2 .

Solution

( 6 k 3 ) −2 Use the Product to a Power Property, ( a b ) m = a m b m . ( 6 ) −2 ( k 3 ) −2 Use the Power Property, ( a m ) n = a m · n . 6 −2 k −6 Use the Definition of a Negative Exponent, a n = 1 a n . 1 6 2 · 1 k 6 Simplify. 1 36 k 6

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Simplify: ( −4 x 4 ) −2 .

1 16 x 8

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Simplify: ( 2 b 3 ) −4 .

1 16 b 12

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Simplify: ( 5 x −3 ) 2 .

Solution

( 5 x −3 ) 2 Use the Product to a Power Property, ( a b ) m = a m b m . 5 2 ( x −3 ) 2 Simplify 5 2 and multiply the exponents of x using the Power Property, ( a m ) n = a m · n . 25 · x −6 Rewrite x −6 by using the Definition of a Negative Exponent, a n = 1 a n . 25 · 1 x 6 Simplify. 25 x 6

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Simplify: ( 8 a −4 ) 2 .

64 a 8

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Simplify: ( 2 c −4 ) 3 .

8 c 12

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To simplify a fraction, we use the Quotient Property and subtract the exponents.

Simplify: r 5 r −4 .

Solution

r 5 r −4 Use the Quotient Property, a m a n = a m n . r 5 ( −4 ) Simplify. r 9

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Questions & Answers

3. When Jenna spent 10 minutes on the elliptical trainer and then did circuit training for20 minutes, her fitness app says she burned 278 calories. When she spent 20 minutes onthe elliptical trainer and 30 minutes circuit training she burned 473 calories. How manycalories does she burn for each minute on the elliptical trainer? How many calories doesshe burn for each minute of circuit training?
Edwin Reply
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
4.5
Ruth
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
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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