If both $a,b\in \mathbb R$ then we have to define $a^b$ by means of complex functions:

\begin{align*} a^b = e^{bLn(a) } = e^{b\ln(|a|)+i b\arg(a)+2\pi pb i} , \end{align*} $p \in \mathbb Z$. Now the exponential function is defined by the series: \begin{align*} e^x = \sum_{n=0}^\infty \frac{x^n}{n!}. \end{align*}

Noting that $\frac{d}{dx}e^x = e^x >0$ (not as easy as it sounds) we know that $f(x) = e^x$ is monotonically increasing on $\mathbb R$. Thus it has an inverse which is called $\ln$. Now this defines $a^b$. Calculating an approximate value of say $(-2)^\pi$ from this is hard (or at least boring).