Wize High School Algebra I Textbook (Common Core) > Exponents & Radicals

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}~} (ba​)n=bnan​ ​
Why does this work?
(ab)n\Large \left(\dfrac{\colorThree{a}}{\colorFive b}\right)^n(ba​)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}}
}=n  timesba​×ba​...×ba​​​

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

Wize Tip
If ab\dfrac{a}{b}ba​ is negative, then
 (ab)n\left(\dfrac{a}{b}\right)^n(ba​)n is positive if nnn is an even exponent
 (ab)n\left(\dfrac{a}{b}\right)^n(ba​)n is negative if nnn 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}~} (ab)n=anbn ​
Why does this work?
(ab)n\Large \left(\colorThree{a}{\colorFive b}\right)^n(ab)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}}
}=n  timesab×ab×...×ab​​

=(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}}=n  times(a×a×...a)​​×n  times(b×b×...b)​​

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

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Example: Exponent Rules for Fractions & Products
Simplify the following.
a) (−23)4\left(\dfrac{-2}{3}\right)^4(3−2​)4

=(−2)4(3)4=\dfrac{(-2)^4}{(3)^4}=(3)4(−2)4​

=+1681=\dfrac{+16}{81}=81+16​ or 1681\dfrac{16}{81}8116​

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

=43x3=4^3x^3=43x3

=64x3=64x^3=64x3

Practice: Power of Fractions
Simplify the following without using a calculator.
a) (25)2\left(\dfrac{2}{5}\right)^2(52​)2

b) (13)3\left(\dfrac{1}{3}\right)^3(31​)3

c) (−34)3\left(-\dfrac{3}{4}\right)^3(−43​)3

d) (−34)2\left(-\dfrac{3}{4}\right)^2(−43​)2

\left(\dfrac{2}{5}\right)^2=

\left(\dfrac{1}{3}\right)^3=

\left(-\dfrac{3}{4}\right)^3=

\left(-\dfrac{3}{4}\right)^2=

I don't know

Practice: Power of Products
Simplify the following without using a calculator.
a) (2x)2\left(2x\right)^2(2x)2

b) (−3y)2\left(-3y\right)^2(−3y)2

c) (−5x)3\left(-5x\right)^3(−5x)3

\left(2x\right)^2=

\left(-3y\right)^2=

\left(-5x\right)^3=

I don't know

Practice: Power of Fractions and Products
Simplify (4yx)2\left(\dfrac{4y}{x}\right)^2(x4y​)2, then evaluate the expression for x=2x=2x=2 and y=12y=\frac{1}{2}y=21​.

What does the expression evaluate to?

I don't know

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 xxxcm  by xxxcm by xxxcm. 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.

a) Enter the volume of Max's cube in cm

^3

, do not include units

b) Enter the volume of Feliks' cube in cm

^3

, do not include units

c) Enter the volume of Sankavi's cube in cm

^3

, do not include units

I don't know