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Simplifying rational exponents

Simplify:

  1. 5 ( 2 x 3 4 ) ( 3 x 1 5 )
  2. ( 16 9 ) 1 2

  1. 30 x 3 4 x 1 5 Multiply the coefficients . 30 x 3 4 + 1 5 Use properties of exponents . 30 x 19 20 Simplify .

  2. ( 9 16 ) 1 2    Use definition of negative exponents . 9 16    Rewrite as a radical . 9 16    Use the quotient rule . 3 4    Simplify .
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Simplify ( 8 x ) 1 3 ( 14 x 6 5 ) .

28 x 23 15

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Access these online resources for additional instruction and practice with radicals and rational exponents.

Key concepts

  • The principal square root of a number a is the nonnegative number that when multiplied by itself equals a . See [link] .
  • If a and b are nonnegative, the square root of the product a b is equal to the product of the square roots of a and b See [link] and [link] .
  • If a and b are nonnegative, the square root of the quotient a b is equal to the quotient of the square roots of a and b See [link] and [link] .
  • We can add and subtract radical expressions if they have the same radicand and the same index. See [link] and [link] .
  • Radical expressions written in simplest form do not contain a radical in the denominator. To eliminate the square root radical from the denominator, multiply both the numerator and the denominator by the conjugate of the denominator. See [link] and [link] .
  • The principal n th root of a is the number with the same sign as a that when raised to the n th power equals a . These roots have the same properties as square roots. See [link] .
  • Radicals can be rewritten as rational exponents and rational exponents can be rewritten as radicals. See [link] and [link] .
  • The properties of exponents apply to rational exponents. See [link] .

Section exercises

Verbal

What does it mean when a radical does not have an index? Is the expression equal to the radicand? Explain.

When there is no index, it is assumed to be 2 or the square root. The expression would only be equal to the radicand if the index were 1.

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Where would radicals come in the order of operations? Explain why.

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Every number will have two square roots. What is the principal square root?

The principal square root is the nonnegative root of the number.

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Can a radical with a negative radicand have a real square root? Why or why not?

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Numeric

For the following exercises, simplify each expression.

Algebraic

For the following exercises, simplify each expression.

8 1 3 x

2 2 + 2 6 x 1 −3 x

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w 3 2 32 w 3 2 50

w 2 w

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12 x 2 + 2 3

3 x 3 x 2

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162 x 6 16 x 4 4

3 2 x 2 4 2

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128 z 3 3 −16 z 3 3

6 z 2 3

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Real-world applications

A guy wire for a suspension bridge runs from the ground diagonally to the top of the closest pylon to make a triangle. We can use the Pythagorean Theorem to find the length of guy wire needed. The square of the distance between the wire on the ground and the pylon on the ground is 90,000 feet. The square of the height of the pylon is 160,000 feet. So the length of the guy wire can be found by evaluating 90,000 + 160,000 . What is the length of the guy wire?

500 feet

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A car accelerates at a rate of 6 4 t  m/s 2 where t is the time in seconds after the car moves from rest. Simplify the expression.

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Extensions

For the following exercises, simplify each expression.

8 16 4 2 2 1 2

−5 2 −6 7

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m n 3 a 2 c −3 a −7 n −2 m 2 c 4

m n c a 9 c m n

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x 64 y + 4 y 128 y

2 2 x + 2 4

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( 250 x 2 100 b 3 ) ( 7 b 125 x )

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64 3 + 256 4 64 + 256

3 3

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Questions & Answers

root under 3-root under 2 by 5 y square
Himanshu Reply
The sum of the first n terms of a certain series is 2^n-1, Show that , this series is Geometric and Find the formula of the n^th
amani Reply
cosA\1+sinA=secA-tanA
Aasik Reply
why two x + seven is equal to nineteen.
Kingsley Reply
The numbers cannot be combined with the x
Othman
2x + 7 =19
humberto
2x +7=19. 2x=19 - 7 2x=12 x=6
Yvonne
because x is 6
SAIDI
what is the best practice that will address the issue on this topic? anyone who can help me. i'm working on my action research.
Melanie Reply
simplify each radical by removing as many factors as possible (a) √75
Jason Reply
how is infinity bidder from undefined?
Karl Reply
what is the value of x in 4x-2+3
Vishal Reply
give the complete question
Shanky
4x=3-2 4x=1 x=1+4 x=5 5x
Olaiya
hi can you give another equation I'd like to solve it
Daniel
what is the value of x in 4x-2+3
Olaiya
if 4x-2+3 = 0 then 4x = 2-3 4x = -1 x = -(1÷4) is the answer.
Jacob
4x-2+3 4x=-3+2 4×=-1 4×/4=-1/4
LUTHO
then x=-1/4
LUTHO
4x-2+3 4x=-3+2 4x=-1 4x÷4=-1÷4 x=-1÷4
LUTHO
A research student is working with a culture of bacteria that doubles in size every twenty minutes. The initial population count was  1350  bacteria. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest whole number, what is the population size after  3  hours?
David Reply
v=lbh calculate the volume if i.l=5cm, b=2cm ,h=3cm
Haidar Reply
Need help with math
Peya
can you help me on this topic of Geometry if l help you
litshani
( cosec Q _ cot Q ) whole spuare = 1_cosQ / 1+cosQ
Aarav Reply
A guy wire for a suspension bridge runs from the ground diagonally to the top of the closest pylon to make a triangle. We can use the Pythagorean Theorem to find the length of guy wire needed. The square of the distance between the wire on the ground and the pylon on the ground is 90,000 feet. The square of the height of the pylon is 160,000 feet. So, the length of the guy wire can be found by evaluating √(90000+160000). What is the length of the guy wire?
Maxwell Reply
the indicated sum of a sequence is known as
Arku Reply
how do I attempted a trig number as a starter
Tumwe Reply
cos 18 ____ sin 72 evaluate
Het Reply
Practice Key Terms 6

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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