I think that an exponent of zero being equal to 1 is an empirical choice that allows a function like a^x to be continuous at x = 0. If you are familiar with infinite power series then you know that the series for e^x yields the value 1 for x = 0; and because all the terms in x vanish at x = 0, the result is exact. This argument is hueristic and lacks rigor but it's the best I can do in my bathrobe.
There are many possible proofs for this, but one that springs to mind is: Take: x^0 And (strangely): 0 = 0 - 0 So we can say: x^0 = x^(0 - 0) From the laws of indices: a^(m - n) = (a^m)/(a^n) We can express: x^(0 - 0) = (x^0)/(x^0) Therefore for any value of x, anything divided by itself is equal to 1. QED Dave