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Calculus/Pure Algebra

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Expert: - 7/9/2010


Question
I am trying to write a formal proof of a rule of exponents.  (ab)^n=a^n*b^n.  It just seems so obvious to me that this is true but I am not sure how to go about proving it.

Thank You,
Sara

Answer
(ab)^n = ab*ab*ab* ... ab*ab with n replicas of a and n replicas of b.

Do to the associative law of mathematics, the order can be rearranged without changing the value.

This gives a*a*a* ... a*a * b*b*b* ... b*b with n a's and n b's.

From here, it can be said that a*a*a* ... a*a with n a's is a^n and similarly for the b's.

Thus, we can say that a*a*a* ... a*a * b*b*b* ... b*b with n a's and n b's is (a^n)(b^n).

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