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Wize High School Grade 9 Math Textbook > Powers & Exponents

Exponent Rule for Power of a Power

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Exponent "Power of a Power" Rule

Whenever we have a power with a base that is also a power, we can use the power of a power rule as a short-cut.

 (am)n=a(m×n) \Large\boxed{~\left(a^{\bcth m}\right)^{\bcfi n}=a^{(\bcth m\times\bcfi{n})}~}

Why does this work?

(am)n\Large{\left(a^{\bcth m}\right)^{\bcfi n}}

=(a×a×...×a⏟m times)n=\Large{ \left( \underbrace{a\times a\times...\times a}_{\bcth m ~\text{times}} \right)^{\bcfi n} }

=(a×a×...×a⏟m times)(a×a×...×a⏟m times)...(a×a×...×a⏟m times)⏟n times=\Large{ \underbrace{\left(\underbrace{a\times a\times...\times a}_{\bcth m ~\text{times}}\right) \left(\underbrace{a\times a\times...\times a}_{\bcth m ~\text{times}}\right)... \left(\underbrace{a\times a\times...\times a}_{\bcth m ~\text{times}}\right)}_{\large{\bcfi n} ~\text{times}} }

=a×a×...×a⏟m×n times=\Large{\underbrace{a\times a\times...\times a}_{\bcth m\times\bcfi n~\text{times}}}

=am+n=\Large{a^{\bcth m+\bcfi n}}
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Example: Exponent Power of a Power Rule

Simplify the following by rewriting them as a single power.

a) (23)7\left(2^3\right)^7

Since we have a power of a power, we can use the power of a power rule:
=23×7=2^{3\times7}
=221=2^{21}


b) (x10)3\left(x^{10}\right)^3

Since we have a power of a power, we can use the power of a power rule:
=x10×3=x^{10\times3}
=x30=x^{30}

Practice: Power of a Power Rule

Select all expressions that have a positive value.

Practice: Rewriting Powers with Different Bases

Express each of the following as powers with the base indicated.
a) 16416^4 with a base of 22
b) 27227^2 with a base of 33
c) 27227^2 with a base of 99

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Example: Extending the Power of a Power Rule

Come up with the general exponent rules to simplify the following.
a) (ambn)p\Large{(a^{\bcth{m}}b^{\bcfi{n}})^{^{\bcf{p}}}}

=(ambn)×(ambn)×...×(ambn)⏟p  times=\underbrace{(a^mb^n)\times(a^mb^n)\times...\times(a^mb^n)}_{p~ \text{ times}}

Rearrange by grouping the aa's together and the bb's together:
=am×am×...×am⏟p  times×bn×bn×...×bn⏟p  times=\underbrace{a^m\times a^m\times...\times a^m}_{p~\text{ times}}\times\underbrace{b^n\times b^n\times...\times b^n}_{p~\text{ times}}
=(am)p×(bn)p=(a^m)^p\times(b^n)^p

Using the power of a power rule:
=(am×p bn×p)=\Large{\boxed{(a^{\bcth m\times \bcf p}~b^{\bcfi n\times\bcf p})}}


b) (ambn)p\Large{\left(\dfrac{a^{\bcth m}}{b^{\bcfi n}}\right)^{\bcf p}}

=(ambn)×(ambn)×...×(ambn)⏟p  times=\underbrace{\left(\dfrac{a^m}{b^n}\right)\times\left(\dfrac{a^m}{b^n}\right)\times...\times\left(\dfrac{a^m}{b^n}\right)}_{p~\text{ times}}

Combine into a single fraction:
=am×am×...×am⏞p  timesbn×bn×...×bn⏟p  times=\dfrac{\overbrace{a^m\times a^m\times...\times a^m}^{p~\text{ times}}}{\underbrace{b^n\times b^n\times...\times b^n}_{p~\text{ times}}}
=(am)p(bn)p=\dfrac{(a^m)^p}{(b^n)^p}

Using the power of a power rule:
=am×pbn×p=\Large{\boxed{\dfrac{a^{\bcth m\times \bcf p}}{b^{\bcfi n\times \bcf p}}}}

Practice: Power of a Power

Simplify the following so that there are no powers of powers.
a) (32x5)4(3^2x^5)^4

b) (−2a542)3\left(-\dfrac{2a^5}{4^2}\right)^3