We all know a^m x a^n = a^(m+n) and it is easy to show why for integer m and n.

How would you prove the law for non-integer m and n?

For rational m and n this would be equivalent to showing

a^(p/q) x a^(r/s) = a^(p/q+r/s) for integer p,q,r and s.

I can do this in the reverse direction, i.e. show that

a^(p/q+r/s) = a^(p/q) + a^(r/s) by combining p/q+r/s as (ps+rq)/qs

and 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?

What about irrationals?

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

Chris Robinson