Using the laws of exponents, we can derive what fractional and
negative powers mean:
$\displaystyle{a^0=a^{1-1} = \frac{a^1}{a^1} = 1,}$ as long as $a \ne 0$.
$\displaystyle{a^{-n}=a^{0-n}=\frac{a^0}{a^n} = \frac{1}{a^n},}$
as long as $a \ne 0$.
$\displaystyle{\left(a^{p/q}\right)^q=a^p}$, so
$\displaystyle{a^{p/q}=\sqrt[q]{a^p}=\left(\sqrt[q]a\right)^p,}$
as long as $a\ge0$.
Notice: the expression $0^0$, as well as negative powers of 0 are not defined, and so are fractional powers of negative numbers, since $a^p$ may not have a $q$-th root.