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?