2.37 Powers $a^{x}$

For $a>0$ we can define $a^{2}=a.a$ by multiplication, $a^{-1}=1/a$ by division and the square root $a^{1/2}$ as the unique $b>0$ such that $b^{2}=a$.

How to calculate

(i) $\pi^{2}$?

(ii) $2^{\pi}$?

Definition For $a>0$ and $x\in{\mathbb{R}}$ we define $a$ to the power $x$ by

This is consistent with earlier notation and has all the usual properties of powers.