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Wize High School Grade 9 Math Textbook > Powers & Exponents

Exponent Rules for Fractions & Products

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Exponent Rules for Fractions & Products

Powers of Fractions

Whenever we have a power where the base is a fraction, we can use this short-cut:
 (ab)n=anbn \Large\boxed{~\left(\dfrac{\colorThree{a}}{\colorFive b}\right)^n=\dfrac{\colorThree a^n}{\colorFive b^n}~}

Why does this work?

(ab)n\Large \left(\dfrac{\colorThree{a}}{\colorFive b}\right)^n

=ab×ab...×ab⏟n  times={ \underbrace{\dfrac{\colorThree a}{\colorFive b}\times \dfrac{\colorThree a}{\colorFive b}...\times \dfrac{\colorThree a}{\colorFive b}}_{n~\text{ times}} }

=anbn=\dfrac{\colorThree a^n}{\colorFive b^n}


Wize Tip
If ab\dfrac{a}{b} is negative, then
  • (ab)n\left(\dfrac{a}{b}\right)^n is positive if nn is an even exponent
  • (ab)n\left(\dfrac{a}{b}\right)^n is negative if nn is an odd exponent

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Powers of Products

Whenever we have a power where the base is a product, we can use this short-cut:
 (ab)n=anbn \Large\boxed{~\left(\colorThree a \colorFive b\right)^n=\colorThree a^n{\colorFive b^n}~}

Why does this work?

(ab)n\Large \left(\colorThree{a}{\colorFive b}\right)^n

=ab×ab×...×ab⏟n  times={ \underbrace{\colorThree a \colorFive b\times \colorThree a \colorFive b\times...\times\colorThree a \colorFive b}_{n~\text{ times}} }

=(a×a×...a)⏟n  times×(b×b×...b)⏟n  times=\underbrace{(\colorThree a\times\colorThree a\times...\colorThree a)}_{n~\text{ times}}\times\underbrace{(\colorFive b\times\colorFive b\times...\colorFive b)}_{n~\text{ times}}

=anbn=\colorThree a^n\colorFive b^n

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Example: Exponent Rules for Fractions & Products

Simplify the following.
a) (−23)4\left(\dfrac{-2}{3}\right)^4

=(−2)4(3)4=\dfrac{(-2)^4}{(3)^4}

=+1681=\dfrac{+16}{81} or 1681\dfrac{16}{81}

b) (4x)3(4x)^3

=43x3=4^3x^3

=64x3=64x^3

Practice: Power of Fractions

Simplify the following without using a calculator.
a) (25)2\left(\dfrac{2}{5}\right)^2

b) (13)3\left(\dfrac{1}{3}\right)^3

c) (−34)3\left(-\dfrac{3}{4}\right)^3

d) (−34)2\left(-\dfrac{3}{4}\right)^2

Practice: Power of Products

Simplify the following without using a calculator.
a) (2x)2\left(2x\right)^2

b) (−3y)2\left(-3y\right)^2

c) (−5x)3\left(-5x\right)^3

Practice: Power of Fractions and Products

Simplify (4yx)2\left(\dfrac{4y}{x}\right)^2, then evaluate the expression for x=2x=2 and y=12y=\frac{1}{2}.

Practice: Rubik's Cube

The Rubik's cube is one of the most well-known puzzle toys in the world. It was first created by a Hungarian design teacher, Erno Rubik in 1974.

Three friends, Max, Feliks, and Sankavi each have a different sized Rubik's cube. Max's cube has dimensions xxcm by xxcm by xxcm. The side length of Feliks' cube is half that of Max's cube. The side length of Sankavi's cube is 1.3 times that of Max's cube.

a) Write a simplified expression for the volume of Max's cube.

b) Write a simplified expression for the volume of Feliks' cube.

c) Write a simplified expression for the volume of Sankavi's cube.