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This leads to the Quotient to a Power Property for Exponents .

Quotient to a power property for exponents

If a and b are real numbers, b 0 , and m is a counting number, then

( a b ) m = a m b m

To raise a fraction to a power, raise the numerator and denominator to that power.

An example with numbers may help you understand this property:

( 2 3 ) 3 = 2 3 3 3 2 3 · 2 3 · 2 3 = 8 27 8 27 = 8 27

Simplify: ( 3 7 ) 2 ( b 3 ) 4 ( k j ) 3 .

Solution


3 sevenths squared.
Use the Quotient Property, ( a b ) m = a m b m . 3 squared divided by 7 squared.
Simplify. 9 forty-ninths.


b thirds to the fourth power.
Use the Quotient Property, ( a b ) m = a m b m . b to the fourth power divided by 3 to the fourth power.
Simplify. b to the fourth power divided by 81.


k divided by j, in parentheses, cubed.
Raise the numerator and denominator to the third power. k cubed divided by j cubed.

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Simplify: ( 5 8 ) 2 ( p 10 ) 4 ( m n ) 7 .

25 64 p 4 10,000 m 7 n 7

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Simplify: ( 1 3 ) 3 ( −2 q ) 3 ( w x ) 4 .

1 27 −8 q 3 w 4 x 4

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Simplify expressions by applying several properties

We’ll now summarize all the properties of exponents so they are all together to refer to as we simplify expressions using several properties. Notice that they are now defined for whole number exponents.

Summary of exponent properties

If a and b are real numbers, and m and n are whole numbers, 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 b m = a m n , a 0 , m > n a m a n = 1 a n m , a 0 , n > m Zero Exponent Definition a o = 1 , a 0 Quotient to a Power Property ( a b ) m = a m b m , b 0

Simplify: ( y 4 ) 2 y 6 .

Solution

( y 4 ) 2 y 6 Multiply the exponents in the numerator. y 8 y 6 Subtract the exponents. y 2

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Simplify: ( m 5 ) 4 m 7 .

m 13

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Simplify: ( k 2 ) 6 k 7 .

k 5

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Simplify: b 12 ( b 2 ) 6 .

Solution

b 12 ( b 2 ) 6 Multiply the exponents in the denominator. b 12 b 12 Subtract the exponents. b 0 Simplify. 1

Notice that after we simplified the denominator in the first step, the numerator and the denominator were equal. So the final value is equal to 1.

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Simplify: n 12 ( n 3 ) 4 .

1

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Simplify: x 15 ( x 3 ) 5 .

1

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Simplify: ( y 9 y 4 ) 2 .

Solution

( y 9 y 4 ) 2 Remember parentheses come before exponents. Notice the bases are the same, so we can simplify inside the parentheses. Subtract the exponents. ( y 5 ) 2 Multiply the exponents. y 10

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Simplify: ( r 5 r 3 ) 4 .

r 8

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Simplify: ( v 6 v 4 ) 3 .

v 6

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Simplify: ( j 2 k 3 ) 4 .

Solution

Here we cannot simplify inside the parentheses first, since the bases are not the same.

( j 2 k 3 ) 4 Raise the numerator and denominator to the third power using the Quotient to a Power Property, ( a b ) m = a m b m . ( j 2 ) 4 ( k 3 ) 4 Use the Power Property and simplify. j 8 k 12

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Simplify: ( a 3 b 2 ) 4 .

a 12 b 8

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Simplify: ( q 7 r 5 ) 3 .

q 21 r 15

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Simplify: ( 2 m 2 5 n ) 4 .

Solution

( 2 m 2 5 n ) 4 Raise the numerator and denominator to the fourth power, using the Quotient to a Power Property, ( a b ) m = a m b m . ( 2 m 2 ) 4 ( 5 n ) 4 Raise each factor to the fourth power. 2 4 ( m 2 ) 4 5 4 n 4 Use the Power Property and simplify. 16 m 8 625 n 4

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Simplify: ( 7 x 3 9 y ) 2 .

49 x 6 81 y 2

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Simplify: ( 3 x 4 7 y ) 2 .

9 x 8 49 y 2

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Simplify: ( x 3 ) 4 ( x 2 ) 5 ( x 6 ) 5 .

Solution

( x 3 ) 4 ( x 2 ) 5 ( x 6 ) 5 Use the Power Property, ( a m ) n = a m · n . ( x 12 ) ( x 10 ) ( x 30 ) Add the exponents in the numerator. x 22 x 30 Use the Quotient Property, a m a n = 1 a n m . 1 x 8

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Simplify: ( a 2 ) 3 ( a 2 ) 4 ( a 4 ) 5 .

1 a 6

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Simplify: ( p 3 ) 4 ( p 5 ) 3 ( p 7 ) 6 .

1 p 15

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Simplify: ( 10 p 3 ) 2 ( 5 p ) 3 ( 2 p 5 ) 4 .

Solution

( 10 p 3 ) 2 ( 5 p ) 3 ( 2 p 5 ) 4 Use the Product to a Power Property, ( a b ) m = a m b m . ( 10 ) 2 ( p 3 ) 2 ( 5 ) 3 ( p ) 3 ( 2 ) 4 ( p 5 ) 4 Use the Power Property, ( a m ) n = a m · n . 100 p 6 125 p 3 · 16 p 20 Add the exponents in the denominator. 100 p 6 125 · 16 p 23 Use the Quotient Property, a m a n = 1 a n m . 100 125 · 16 p 17 Simplify. 1 20 p 17

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Simplify: ( 3 r 3 ) 2 ( r 3 ) 7 ( r 3 ) 3 .

9 r 18

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Questions & Answers

how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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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
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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?
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Mayra has $124 in her checking account. She writes a check for $152. What is the New Balance in her checking account?
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ashley
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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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