\begin{definition}[Definition:Power (Algebra)/Power of Zero]
Let $r \in \R$ be a real number.
(This includes the situation where $r \in \Z$ or $r \in \Q$.)
When $x=0$, $x^r$ is defined as follows:
:$0^r = \begin{cases}
1 & : r = 0 \\
0 & : r > 0 \\
\text{Undefined} & : r < 0 \\
\end{cases}$
This takes account of the awkward case $0^0$: it is "generally accepted" that $0^0 = 1$ as this convention agrees with certain general results which would otherwise need a special case.
\end{definition}