Here's a question that's bothered me ever since highschool, and I've never heard a good answer.
I know that mathematicians can define operators to mean whatever they want, as long as their system of mathematics is self-consistent. But some definitions are less useful than others when one wishes to use math to model our physical world.
And so my question regards the soundness of using non-integer exponents when modeling physical systems. I accept the validity of positive-integer exponent rules, such as $a^n \times a^m = a^{n+m}, n \in I, m \in I$, because this degenerates into straight-forward multiplication. And that, in turn, maps pretty well to grids of apples on a picnic blanket.
But why do people so readily (or at least without public discussion) accept that $a^x \times a^y = a^{x+y}$ even when $x$ and $y$ are fractions, irrationals, or even complex numbers? More precisely, why do people who are using equations to model physical systems not demand a justification for the validity of such a definition of exponentiation, before being willing to apply it in the manipulation of their mathematical physical models?