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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

help me understand graphs
Marlene Reply
what kind of graphs?
bruce
function f(x) to find each value
Marlene
I am in algebra 1. Can anyone give me any ideas to help me learn this stuff. Teacher and tutor not helping much.
Marlene
Given f(x)=2x+2, find f(2) so you replace the x with the 2, f(2)=2(2)+2, which is f(2)=6
Melissa
if they say find f(5) then the answer would be f(5)=12
Melissa
I need you to help me Melissa. Wish I can show you my homework
Marlene
How is f(1) =0 I am really confused
Marlene
-65r to the 4th power-50r cubed-15r squared+8r+23 ÷ 5r
WENDY Reply
write in this form a/b answer should be in the simplest form 5%
August Reply
convert to decimal 9/11
August
Equation in the form of a pending point y+2=1/6(×-4)
Jose Reply
write in simplest form 3 4/2
August
definition of quadratic formula
Ahmed Reply
From Google: The quadratic formula, , is used in algebra to solve quadratic equations (polynomial equations of the second degree). The general form of a quadratic equation is , where x represents a variable, and a, b, and c are constants, with . A quadratic equation has two solutions, called roots.
Melissa
what is the answer of w-2.6=7.55
What Reply
10.15
Michael
w = 10.15 You add 2.6 to both sides and then solve for w (-2.6 zeros out on the left and leaves you with w= 7.55 + 2.6)
Korin
Nataly is considering two job offers. The first job would pay her $83,000 per year. The second would pay her $66,500 plus 15% of her total sales. What would her total sales need to be for her salary on the second offer be higher than the first?
Mckenzie Reply
x > $110,000
bruce
greater than $110,000
Michael
Estelle is making 30 pounds of fruit salad from strawberries and blueberries. Strawberries cost $1.80 per pound, and blueberries cost $4.50 per pound. If Estelle wants the fruit salad to cost her $2.52 per pound, how many pounds of each berry should she use?
nawal Reply
$1.38 worth of strawberries + $1.14 worth of blueberries which= $2.52
Leitha
how
Zaione
is it right😊
Leitha
lol maybe
Robinson
8 pound of blueberries and 22 pounds of strawberries
Melissa
8 pounds x 4.5 = 36 22 pounds x 1.80 = 39.60 36 + 39.60 = 75.60 75.60 / 30 = average 2.52 per pound
Melissa
8 pounds x 4.5 equal 36 22 pounds x 1.80 equal 39.60 36 + 39.60 equal 75.60 75.60 / 30 equal average 2.52 per pound
Melissa
hmmmm...... ?
Robinson
8 pounds x 4.5 = 36 22 pounds x 1.80 = 39.60 36 + 39.60 = 75.60 75.60 / 30 = average 2.52 per pound
Melissa
The question asks how many pounds of each in order for her to have an average cost of $2.52. She needs 30 lb in all so 30 pounds times $2.52 equals $75.60. that's how much money she is spending on the fruit. That means she would need 8 pounds of blueberries and 22 lbs of strawberries to equal 75.60
Melissa
good
Robinson
👍
Leitha
thanks Melissa.
Leitha
nawal let's do another😊
Leitha
we can't use emojis...I see now
Leitha
Sorry for the multi post. My phone glitches.
Melissa
Vina has $4.70 in quarters, dimes and nickels in her purse. She has eight more dimes than quarters and six more nickels than quarters. How many of each coin does she have?
Mckenzie Reply
10 quarters 16 dimes 12 nickels
Leitha
A private jet can fly 1,210 miles against a 25 mph headwind in the same amount of time it can fly 1,694 miles with a 25 mph tailwind. Find the speed of the jet.
Crispy Reply
wtf. is a tail wind or headwind?
Robert
48 miles per hour with headwind and 68 miles per hour with tailwind
Leitha
average speed is 58 mph
Leitha
Into the wind (headwind), 125 mph; with wind (tailwind), 175 mph. Use time (t) = distance (d) ÷ rate (r). since t is equal both problems, then 1210/(x-25) = 1694/(×+25). solve for x gives x=150.
bruce
the jet will fly 9.68 hours to cover either distance
bruce
Riley is planning to plant a lawn in his yard. He will need 9 pounds of grass seed. He wants to mix Bermuda seed that costs $4.80 per pound with Fescue seed that costs $3.50 per pound. How much of each seed should he buy so that the overall cost will be $4.02 per pound?
Vonna Reply
33.336
Robinson
Amber wants to put tiles on the backsplash of her kitchen counters. She will need 36 square feet of tiles. She will use basic tiles that cost $8 per square foot and decorator tiles that cost $20 per square foot. How many square feet of each tile should she use so that the overall cost of the backsplash will be $10 per square foot?
Imaan Reply
Ivan has $8.75 in nickels and quarters in his desk drawer. The number of nickels is twice the number of quarters. How many coins of each type does he have?
mikayla Reply
2q=n ((2q).05) + ((q).25) = 8.75 .1q + .25q = 8.75 .35q = 8.75 q = 25 quarters 2(q) 2 (25) = 50 nickles Answer check 25 x .25 = 6.25 50 x .05 = 2.50 6.25 + 2.50 = 8.75
Melissa
John has $175 in $5 and $10 bills in his drawer. The number of $5 bills is three times the number of $10 bills. How many of each are in the drawer?
mikayla Reply
7-$10 21-$5
Robert
Enrique borrowed $23,500 to buy a car. He pays his uncle 2% interest on the $4,500 he borrowed from him, and he pays the bank 11.5% interest on the rest. What average interest rate does he pay on the total $23,500? (Round your answer to the nearest tenth of a percent.)
Parker Reply
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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