Suppose you have three finite sets A, B and C, each with a, b and c elements. Let A->B be the set of functions from A to B. Then it has b^a elements. Let AxB

Suppose you have three finite sets A, B and C, each with a, b and c elements. Let A->B be the set of functions from A to B. Then it has b^a elements. Let AxB the set of couples with the first element in A and the second on B. Then it has ab elements. So to prove that (a^b)^c = a^(cb) you have to prove that there is a bijection between C->(B->A) and (CxB)->A. Ever heard of currying and uncurrying? (BTW, I didn't really use the hypothesis that the sets are finite, the proof is also valid for transfinites if you care about those)