Why do negative exponents work the way they do? Why is a value with a negative exponent equal to the multiplicative inverse but with a positive exponent?
$$a^{-b} = \frac{1}{a^b}$$

Think of it this way: exponentiation is equivalent to repeated multiplication, in the sense that, for example, $3^4=3\times3\times3\times3$; so a multiplication repeated a negative number of times should use the multiplicative inverse, division. Therefore, a negative exponentiation could be represented as a repeated division, which would be equivalent to $a^{-b}=\frac{1}{a^b}$.
