Well, youmean"a^(p/q+ r/s)= a^(p/q) x a^{r/s}", not "+".

What do you mean by "in the forward direction". You have shown thatand using the fact that ps and rq are integers and using the law for integers.

But how do you show the law is true working from a proof in the forward direction?

a^(p/q+ r/s)= a^(p/q)+ a^{r}{s}. There is no "direction" in "="- it doesn't matter on which side you did the calculation, they are equal.

I would point out that this (or any proof that a^(x+y)= a^x a^y is "circular". We really define a^(1/n) in such a way as tomakethis true.

How do youWhat about irrationals?

How would you show that a^pi x a^e = a^(pi+e) say?

Chris Robinsondefinea^e? Usually we define exponential powers "by continuity". That is if is a sequence of rational numbers converging to x, we define to be limit of the sequence . That would, of course, require showing that sequencedoesconverge and, for any sequence of rational numbers converging to x, that sequence will converge to the same thing, but that's not too difficult.

From that, let be a sequence of rational numbers converging to the irrational number x and let be a sequence of rational numbers converging to the irrational number y. Then converges to . But each term in that sequence is the same as because all and and that sequence converges to .