TITLE: Why do negative exponents work the way they do?
QUESTION [27 upvotes]: Why is a value with a negative exponent equal to the multiplicative inverse but with a positive exponent?
$$a^{-b} = \frac{1}{a^b}$$

REPLY [5 votes]: (The OP doesn't seem to understand what a negative exponent means. Assuming this was not actually the question asked for homework. Something like this should have been covered by his teacher.)
Just follow the pattern. For non-zero $x$, we have
$x^3=x\cdot x \cdot x$
$x^2=x\cdot x$
$x^1=x$
$x^0=1$
Continue dividing by $x$ each time, as above...
$x^{-1}=1\div x =\frac{1}{x}$
$x^{-2}=\frac{1}{x}\div x =\frac{1}{x^2}$
$x^{-3}=\frac{1}{x^2}\div x =\frac{1}{x^3}$
and so on.