Proof real to the power of real

Question

Let a be a real number and p a non zero real number. Then $a^p$ is also a real number. What definitions, propositions and etc are required to prove this?

Answer 1

The answer to your question depends on where you want to begin. If you accept the real numbers and elementary calculus as given one approach is to define $e^x$ using appropriate tools - perhaps the power series. Then define the natural logarithm and, finally, $$ a^p = \exp(p \ln a) . $$ If you don't want to use the power series you can start by defining the natural logarithm as an integral, then the exponential function as its inverse.

This works for $a > 0$. For negative $a$ things get more complicated. The same final formula works, but the logarithm is not well defined. You need complex analysis to understand the way in which it is multivalued.

Answer 2

Obviously, you must defined first the meaning of $a^p$, when $a,p\in\mathbb R$ and $p\neq0$. But I have never seen a definition which defines, say, $(-1)^{\sqrt2}$.