Why is the math for negative exponents so? This is what we are taught:
$$5^{-2} = \left({\frac{1}{5}}\right)^{2}$$
but I don't understand why we take the inverse of the base when we have a negative exponent.  Can anyone explain why?

$$
\begin{align}
a\cdot 3^{10} & = a\cdot 3\cdot 3\cdot 3\cdot 3\cdot 3\cdot 3\cdot 3\cdot 3\cdot 3\cdot 3 \\  \\
a\cdot 3^6 & = a\cdot 3\cdot 3\cdot 3\cdot 3\cdot 3\cdot 3
\end{align}
$$
To go from the second line to the first, multiply by 3 four times.
To go from the first to the second, multiply by 3 minus four times.
$10 = 6 + 4$
$6 = 10 + (-4)$
$$
\begin{align}
a\cdot 3^{10} & = a\cdot 3^{6+4} \\  \\
a\cdot 3^6 & = a\cdot 3^{10 + (-4)}
\end{align}
$$
