I've written this before but I have nothing better to do!
You probably saw "\(\displaystyle a^n\)" for n a positive integer as "a multiplied by itself n times". From that definition we can get two important properties:
\(\displaystyle (a^n)(a^m)= a^{n+ m}\) and \(\displaystyle (a^n)^m= a^{mn}\).
We can see the first by thinking of \(\displaystyle a^{n+ m}\) as the product of n+ m copies of a multiplied together. We can separate those into n copies and m copies so \(\displaystyle a^n\) and \(\displaystyle a^m\).
For \(\displaystyle (a^n)^m\) think of this as m copies of \(\displaystyle a^n\) and think of each \(\displaystyle a^n\) as n copies of a multiplied together on one row, m rows. That is a total of mn copies of a multiplied together.
Okay, what in the world could we mean by "\(\displaystyle a^0\)"? We can't multiply "0 copies" of a together! So we need to define \(\displaystyle a^0\) separately. We are free to define it any way we want but it would be nice if \(\displaystyle (a^n)(a^m)= a^{n+ m}\) were true even when n= 0. So we want \(\displaystyle (a^n)(a^0)= a^{n+ 0}\). But 0 is the "additive identity"- \(\displaystyle m+ 0= m\) so that says \(\displaystyle (a^n)(a^0)= a^n\). As long as \(\displaystyle a^n\ne 0\), which means \(\displaystyle a\ne 0\), we can divide both sides by \(\displaystyle a^n\): \(\displaystyle a^0= 1\). That is, in order to have \(\displaystyle (a^n)(a^m)= a^{n+m}\) we must define \(\displaystyle a^0= 1\). Again, that is if \(\displaystyle a\ne 0\). "\(\displaystyle 0^0\)" is NOT defined.
What about negative integers? How should we define \(\displaystyle a^{-n}\)? Again, we would like \(\displaystyle (a^n)(a^m)= a^{n+ m}\) for m= -n. In that case, we have \(\displaystyle (a^n)(a^{-n})= a^{n- n}= a^0= 1\). As long as a is not 0, we can divide by \(\displaystyle a^n\) to get \(\displaystyle a^{-n}= \frac{1}{a^n}\).
That is, for \(\displaystyle a\ne 0\), \(\displaystyle a^{-n}= \frac{1}{a^n}\). In particular, \(\displaystyle 8^{-1/2}= \frac{1}{8^{1/2}}\).