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Simplify: ( 16 m 1 3 ) 3 2 ( 81 n 2 5 ) 3 2 .

64 m 1 2 729 n 3 5

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Simplify: ( m 3 n 9 ) 1 3 ( p 4 q 8 ) 1 4 .

Solution


  1. ( m 3 n 9 ) 1 3 First we use the Product to a Power Property. ( m 3 ) 1 3 ( n 9 ) 1 3 To raise a power to a power, we multiply the exponents. m n 3


  2. ( p 4 q 8 ) 1 4 First we use the Product to a Power Property. ( p 4 ) 1 4 ( q 8 ) 1 4 To raise a power to a power, we multiply the exponents. p q 2
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We will use both the Product and Quotient Properties in the next example.

Simplify: x 3 4 · x 1 4 x 6 4 y 4 3 · y y 2 3 .

Solution


  1. x 3 4 · x 1 4 x 6 4 Use the Product Property in the numerator, add the exponents. x 2 4 x 6 4 Use the Quotient Property, subtract the exponents. x 8 4 Simplify. x 2


  2. y 4 3 · y y 2 3 Use the Product Property in the numerator, add the exponents. y 7 3 y 2 3 Use the Quotient Property, subtract the exponents. y 9 3 Simplify. y 3
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Simplify: m 2 3 · m 1 3 m 5 3 n 1 6 · n n 11 6 .

m 2 n 3

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Simplify: u 4 5 · u 2 5 u 13 5 v 1 2 · v v 7 2 .

u 3 v 5

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Key concepts

  • Summary of Exponent Properties
  • If a , b are real numbers and m , n are rational 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 a n = a m n , a 0 , m > n
      a m a n = 1 a n m , a 0 , n > m
    • Zero Exponent Definition a 0 = 1 , a 0
    • Quotient to a Power Property ( a b ) m = a m b m , b 0

Practice makes perfect

Simplify Expressions with a 1 n

In the following exercises, write as a radical expression.


x 1 2
y 1 3
z 1 4

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r 1 2
s 1 3
t 1 4

r s 3 t 4

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u 1 5
v 1 9
w 1 20

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g 1 7
h 1 5
j 1 25

g 7 h 5 j 25

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In the following exercises, write with a rational exponent.


x 7
y 9
f 5

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r 8
.
t 4

r 1 8 s 1 10 t 1 4

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u 5
v
.

u 1 5 v 1 2 w 1 16

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7 c 3
12 d 7
3 5 f 4

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5 x 4
9 y 8
7 3 z 5

( 5 x ) 1 4 ( 9 y ) 1 8 7 ( 3 z ) 1 5

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21 p
8 q 4
4 36 r 6

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25 a 3
3 b
.

( 25 a ) 1 3 ( 3 b ) 1 2 ( 40 c ) 1 10

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In the following exercises, simplify.


81 1 2
125 1 3
64 1 2

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625 1 4
243 1 5
32 1 5

5 3 2

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16 1 4
16 1 2
3125 1 5

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216 1 3
32 1 5
81 1 4

6 2 3

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( −216 ) 1 3
216 1 3
( 216 ) 1 3

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( −243 ) 1 5
243 1 5
( 243 ) 1 5

−3 −3 1 3

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( −1 ) 1 3
−1 1 3
( 1 ) 1 3

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( −1000 ) 1 3
1000 1 3
( 1000 ) 1 3

−10 −10 1 10

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( −81 ) 1 4
81 1 4
( 81 ) 1 4

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( −49 ) 1 2
49 1 2
( 49 ) 1 2

not a real number −7 1 7

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( −36 ) 1 2
36 1 2
( 36 ) 1 2

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( −1 ) 1 4
( 1 ) 1 4
1 1 4

not a real number 1 −1

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( −100 ) 1 2
100 1 2
( 100 ) 1 2

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( −32 ) 1 5
( 243 ) 1 5
125 1 3

−2 1 3
−5

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Simplify Expressions with a m n

In the following exercises, write with a rational exponent.


m 5
n 2 3
p 3 4

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r 7 4
s 3 5
t 7 3

r 7 4 s 3 5 t 7 3

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u 2 5
v 8 5
w 4 9

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a 3
b 5
c 5 3

a 1 3 b 1 5 c 5 3

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In the following exercises, simplify.


16 3 2
8 2 3
10,000 3 4

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1000 2 3
25 3 2
32 3 5

100 125 8

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27 5 3
16 5 4
32 2 5

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16 3 2
125 5 3
64 4 3

64 3125 256

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32 2 5
27 2 3
25 3 2

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64 5 2
81 3 2
27 4 3

32,768 1 729 1 81

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25 3 2
9 3 2
( −64 ) 2 3

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100 3 2
49 5 2
( −100 ) 3 2

1000 1 16,807 not a real number

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9 3 2
9 3 2
( −9 ) 3 2

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64 3 2
64 3 2
( −64 ) 3 2

−512
1 512 not a real number

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100 3 2
100 3 2
( −100 ) 3 2

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49 3 2
49 3 2
( −49 ) 3 2

−343 1 343 not a real number

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Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify.


4 5 8 · 4 11 8
m 7 12 · m 17 12
p 3 7 · p 18 7

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6 5 2 · 6 1 2
n 2 10 · n 8 10
q 2 5 · q 13 5

216 n q 3

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5 1 2 · 5 7 2
c 3 4 · c 9 4
d 3 5 · d 2 5

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10 1 3 · 10 5 3
x 5 6 · x 7 6
y 11 8 · y 21 8

100 x 2 y 4

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( m 6 ) 5 2
( n 9 ) 4 3
( p 12 ) 3 4

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( a 12 ) 1 6
( b 15 ) 3 5
( c 11 ) 1 11

a 2 b 9
c

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( x 12 ) 2 3
( y 20 ) 2 5
( z 16 ) 1 16

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( h 6 ) 4 3
( k 12 ) 3 4
( j 10 ) 7 5

h 8 k 9 j 14

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x 7 2 x 5 2
y 5 2 y 1 2
r 4 5 r 9 5

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s 11 5 s 6 5
z 7 3 z 1 3
w 2 7 w 9 7

s z 2 1 w

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t 12 5 t 7 5
x 3 2 x 1 2
m 13 8 m 5 8

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u 13 9 u 4 9
r 15 7 r 8 7
n 3 5 n 8 5

u r 1 n

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( 9 p 2 3 ) 5 2
( 27 q 3 2 ) 4 3

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( 81 r 4 5 ) 1 4
( 64 s 3 7 ) 1 6

3 r 1 5 2 s 1 14

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( 16 u 1 3 ) 3 4
( 100 v 2 5 ) 3 2

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( 27 m 3 4 ) 2 3
( 625 n 8 3 ) 3 4

9 m 1 2 125 n 2

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Questions & Answers

how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
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
Hi everyone, I'm glad to be connected with you all. from France.
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
Mayra has $124 in her checking account. She writes a check for $152. What is the New Balance in her checking account?
REVOLUTION Reply
-28$
ashley
-$28
Stephanie
Practice Key Terms 1

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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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