Indeed this is informal and could be made more rigorous, but even at the highest level of rigor, I think it's most natural to do integral exponents and then rational. Indeed you have to construct the integers before you can construct the rationals.
That would be difficult, since a formal construction of even the real numbers is a somewhat advanced (3rd or 4th year college mathematics) topic. I forget the details, but I believe a^n for real a and complex n is formally defined using the exponential function (e^x).
