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?

We know that positive exponents add, e.g. $5^3 \times 5^2 = 5^5$.  If you accept that $5^0 = 1$, then it makes sense that $5^{-2} \times 5^2 = 5^0 = 1$.  That means that $5^{-2} = \frac{1}{5^2}$.
