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Simplify: 3 x 5 + 3 x 5 3 9 3 9 3 .

2 3 x 5 2 9 3

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Simplify: 10 y 4 + 10 y 4 5 32 6 3 32 6 .

2 10 y 4 2 32 6

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When an expression does not appear to have like radicals, we will simplify each radical first. Sometimes this leads to an expression with like radicals.

Simplify: 54 3 16 3 48 4 + 243 4 .

Solution


  1. 54 3 16 3 Rewrite each radicand using perfect cube factors. 27 3 · 2 3 8 3 · 2 3 Rewrite the perfect cubes. ( 3 ) 3 3 2 3 ( 2 ) 3 3 2 3 Simplify the radicals where possible. 3 2 3 2 2 3 Combine like radicals. 2 3


  2. 48 4 + 243 4 Rewrite using perfect fourth power factors. 16 4 · 3 4 + 81 4 · 3 4 Rewrite the perfect fourth powers. ( 2 ) 4 4 3 4 + ( 3 ) 4 4 3 4 Simplify the radicals where possible. 2 3 4 + 3 3 4 Combine like radicals. 5 3 4
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Simplify: 192 3 81 3 32 4 + 512 4 .

3 3 6 2 4

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Simplify: 108 3 250 3 64 5 + 486 5 .

2 3 5 2 5

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Simplify: 24 x 4 3 −81 x 7 3 162 y 9 4 + 516 y 5 4 .

Solution


  1. 24 x 4 3 −81 x 7 3 Rewrite each radicand using perfect cube factors. 8 x 3 3 · 3 x 3 −27 x 6 3 · 3 x 3 Rewrite the perfect cubes. ( 2 x ) 3 3 3 x 3 ( −3 x 2 ) 3 3 3 x 3 Simplify the radicals where possible. 2 x 3 x 3 ( −3 x 2 3 x 3 )


  2. 162 y 9 4 + 516 y 5 4 Rewrite each radicand using perfect fourth power factors. 81 y 8 4 · 2 y 4 + 256 y 4 4 · 2 y 4 Rewrite the perfect fourth powers. ( 3 y 2 ) 4 4 · 2 y 4 + ( 4 y ) 4 4 · 2 y 4 Simplify the radicals where possible. 3 y 2 2 y 4 + 4 | y | 2 y 4
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Simplify: 32 y 5 3 −108 y 8 3 243 r 11 4 + 768 r 10 4 .

2 y 4 y 2 3 + 3 y 2 4 y 2 3 3 r 2 3 r 3 4 + 4 r 2 3 r 2 4

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Simplify: 40 z 7 3 −135 z 4 3 80 s 13 4 + 1280 s 6 4 .

2 z 2 5 z 3 + 3 z 5 z 3 2 | s 3 | 5 s 4 + 4 | s | 5 s 4

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Access these online resources for additional instruction and practice with simplifying higher roots.

Key concepts

  • Properties of
  • a n when n is an even number and
    • a 0 , then a n is a real number
    • a < 0 , then a n is not a real number
    • When n is an odd number, a n is a real number for all values of a .
    • For any integer n 2 , when n is odd a n n = a
    • For any integer n 2 , when n is even a n n = | a |
  • a n is considered simplified if a has no factors of m n .
  • Product Property of n th Roots
    a b n = a n · b n and a n · b n = a b n
  • Quotient Property of n th Roots
    a b n = a n b n and a n b n = a b n
  • To combine like radicals, simply add or subtract the coefficients while keeping the radical the same.

Practice makes perfect

Simplify Expressions with Higher Roots

In the following exercises, simplify.


216 3
256 4
32 5

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27 3
16 4
243 5

3 2 3

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512 3
81 4
1 5

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125 3
1296 4
1024 5

5 6 4

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−8 3
−81 4
−32 5

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−64 3
−16 4
−243 5

−4 not real −3

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−125 3
−1296 4
−1024 5

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−512 3
−81 4
−1 5

−8 not a real number −1

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a 3 3
.

a | b |

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k 8 8
p 6 6

| k | | p |

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a 10 5
b 27 3

a 2 b 9

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r 12 6
s 30 3

r 2 s 10

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16 x 8 4
64 y 12 6

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−8 c 9 3
125 d 15 3

−2 c 3 5 d 5

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216 a 6 3
32 b 20 5

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128 r 14 7
81 s 24 4

2 r 2 3 s 6

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Use the Product Property to Simplify Expressions with Higher Roots

In the following exercises, simplify.

u 7 5 v 11 6

u u 2 5 v v 5 6

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p 8 5 q 8 3

p p 3 5 q 2 q 2 3

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625 3 128 6

5 5 3 2 2 6

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3125 4 81 3

5 5 4 3 3 3

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108 x 5 3 48 y 6 4

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96 a 7 5 375 b 4 3

2 a 3 a 2 5 5 b 3 b 3

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405 m 10 4 160 n 8 5

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512 p 5 3 324 q 7 4

8 p p 2 3 3 q 4 q 3 4

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−864 3 −256 4

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−486 5 −64 6

−3 2 5 not real

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−8 3 −16 4

−2 not real

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Use the Quotient Property to Simplify Expressions with Higher Roots

In the following exercises, simplify.

p 11 p 2 3 q 17 q 13 4

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d 12 d 7 5 m 12 m 4 8

d | m |

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u 21 u 11 5 v 30 v 12 6

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r 14 r 5 3 c 21 c 9 4

r 2 | c 3 |

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64 4 2 4 128 x 8 5 2 x 2 5

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−625 3 5 3 80 m 7 4 5 m 4

−5 4 m m 2 4

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1050 2 3 486 y 9 2 y 3 4

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162 6 3 160 r 10 5 r 3 4

3 6 3 2 | r | 2 r 3 4

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54 a 8 b 3 3 64 c 5 d 2 4

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96 r 11 s 3 5 128 u 7 v 3 6

2 r 2 3 r 5 s 3 2 u 3 2 u v 3 6 v

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81 s 8 t 3 3 64 p 15 q 12 4

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625 u 10 v 3 3 729 c 21 d 8 4

5 u 3 5 u 3 v 3 c 5 9 c 4 d 2

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Add and Subtract Higher Roots

In the following exercises, simplify.


8 p 7 + 8 p 7
3 25 3 25 3

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15 q 3 + 15 q 3
2 27 4 6 27 4

2 15 q 3 −4 27 4

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3 9 x 5 + 7 9 x 5
8 3 q 7 2 3 q 7

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81 3 192 3
512 4 32 4

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250 3 54 3
243 4 1875 4

5 5 3 3 2 3 −2 3 4

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128 3 + 250 3
729 5 + 96 5

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243 4 + 1250 4
2000 3 + 54 3

3 3 4 + 5 2 4 13 2 3

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64 a 10 3 −216 a 12 3
486 u 7 4 + 768 u 3 4

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80 b 5 3 −270 b 3 3
160 v 10 4 1280 v 3 4

2 b 10 b 2 3 + 3 b 10 3 2 v 2 10 v 2 4 4 5 v 3 4

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Mixed Practice

In the following exercises, simplify.

128 x 8 5 2 x 2 5

2 x 2 x 5

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128 u 7 v 3 6

2 u 3 2 u v 3 6 v

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64 a 10 3 −216 a 12 3

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486 u 7 4 + 768 u 3 4

3 u 6 u 3 4 + 4 3 u 3 4

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Everyday math

Population growth The expression 10 · x n models the growth of a mold population after n generations. There were 10 spores at the start, and each had x offspring. So 10 · x n is the number of offspring at the fifth generation. At the fifth generation there were 10,240 offspring. Simplify the expression 10,240 10 5 to determine the number of offspring of each spore.

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Spread of a virus The expression 3 · x n models the spread of a virus after n cycles. There were three people originally infected with the virus, and each of them infected x people. So 3 · x 4 is the number of people infected on the fourth cycle. At the fourth cycle 1875 people were infected. Simplify the expression 1875 3 4 to determine the number of people each person infected.

5

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Writing exercises

Explain how you know that x 10 5 = x 2 .

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Explain why −64 4 is not a real number but −64 3 is.

Answers may vary.

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Self check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

This table has four columns and five rows. The first row labels each column: “I can…,” “Confidentaly,” “With some help,” and “No – I don’t get it!” The rows under the “I can…,” column read, “simplify expressions with hither roots.,” “use the product property to simplify expressions with higher roots.,” “use the quotient property to simplify expressions with higher roots.,” and “add and subtract higher roots.” The rest of the rows under the columns are empty.

What does this checklist tell you about your mastery of this section? What steps will you take to improve?

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 4

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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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