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Wize High School Grade 9 Math Textbook > Powers & Exponents

Product Rule for Multiplying Powers

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Exponent Product Rule for Multiplying Powers

Whenever we multiply powers with the same base, we can use the exponent product rule as a short-cut.

 am×an=am+n \Large\boxed{~a^{\bcth{m}}\times a^{\bcfi{n}}=a^{{\bcth{m}}+{\bcfi{n}}}~}

Why does this work?

am×an\Large{a^{\bcth{m}}\times a^{\bcfi{n}}}

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

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

=am+n=\Large{a^{\bcth m+\bcfi{n}}}


Watch Out!
The exponent product rule does NOT work when you're trying to multiply powers with different bases or when you're trying to add powers with the same bases.

Common mistakes to avoid:
  • am×bn≠(ab)m+na^m\times b^n\neq (ab)^{m+n}
  • am+an≠am+na^m+a^n\neq a^{m+n}

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Example: Product Rule for Multiplying Powers

Simplify the following by rewriting them as a single power.

a) 23×272^3\times2^7

Since we are multiplying two powers with the same base 22, we can use the exponent product rule:
=23+7=2^{3+7}
=210=2^{10}


b) x3×x10x^3\times x^{10}

Since we are multiplying two powers with the same base xx, we can use the exponent product rule:
=x3+10=x^{3+10}
=x13=x^{13}

c) (13)2(13)5\left(\dfrac{1}{3}\right)^2\left(\dfrac{1}{3}\right)^5

Since we are multiplying two powers with the same base 13\dfrac{1}{3}, we can use the exponent product rule:
=(13)2+5=\left(\dfrac{1}{3}\right)^{2+5}
=(13)7=\left(\dfrac{1}{3}\right)^7

Practice: Product Rule for Multiplying Powers


Simplify the following.
a) (x5)(w2)(x4)(x3)(w2)(x^5)(w^2)(x^4)(x^3)(w^2)
b) −4×(a3)(a4)(−4)5-4\times(a^3)(a^4)(-4)^5

Practice: Product Rule for Multiplying Powers

Select all of the true statements.

(Pick all of the equations that are correct)