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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

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
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A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
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A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
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2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
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you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
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can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
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Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
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progressive wave
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A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
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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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