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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 many typos can we find...?
Chris Reply
5
Joseph
In the LCM Prime Factors exercises, the LCM of 28 and 40 is 280. Not 420!
Chris Reply
4x+7y=29,x+3y=11 substitute method of linear equation
Srinu Reply
substitute method of linear equation
Srinu
Solve one equation for one variable. Using the 2nd equation, x=11-3y. Substitute that for x in first equation. this will find y. then use the value for y to find the value for x.
bruce
I want to learn
Elizebeth
help
Elizebeth
I want to learn. Please teach me?
Wayne
1) Use any equation, and solve for any of the variables. Since the coefficient of x (the number in front of the x) in the second equation is 1 (it actually isn't shown, but 1 * x = x), use that equation. Subtract 3y from both sides (this isolates the x on the left side of the equal sign).
bruce
2) This results in x=11-3y. x is note in terms of y. Use that as the value of x and substitute for all x in the first equation. The first equation becomes 4(11-3y)+7y =29. Note that the only variable left in the first equation is the y. If you have multiple variable, then something is wrong.
bruce
3) Distribute (multiply) the 4 across 11-3y to get 44-12y. Add this to the 7y. So, the equation is now 44-5y=29.
bruce
4) Solve 44-5y=29 for y. Isolate the y by subtracting 44 from birth sides, resulting in -5y=-15. Now, divide birth sides by -5 (since you have -5y). This results in y=3. You now have the value of one variable.
bruce
5) The last step is to take the value of y from Step 4) and substitute into the 2nd equation. Therefore: x+3y=11 becomes x+3(3)=11. Then multiplying, x+9=11. Finally, solve for x by subtracting 9 from both sides. Therefore, x=2.
bruce
6) The ordered pair of (2, 3) is the proposed solution. To check, substitute those values into either equation. If the result is true, then the solution is correct. 4(2)+7(3)=8+21=29. TRUE! Finished.
bruce
At 1:30 Marlon left his house to go to the beach, a distance of 5.625 miles. He rose his skateboard until 2:15, and then walked the rest of the way. He arrived at the beach at 3:00. Marlon's speed on his skateboard is 1.5 times his walking speed. Find his speed when skateboarding and when walking.
Andrew Reply
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
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
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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