Actually one of the ways to prove that requires the fact that R completes Q. The concept of completeness was introduced by Cauchy in the 19th century and it might not have been completely popularised in 1869.
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).
At least in the Netherlands, construction of the reals (for example from rational Cauchy sequences) is standard 1st semester stuff. Understanding reals is required or provides a good source of examples for virtually all mathematics courses, so I can't imagine how some universities teach mathematics without it.
