Negative Exponents

What should $a^{-1}$ be? Multiplying it by $a$ gives

$$a^{-1} \cdot a = a^{-1} a^1 = a^{-1+1} = a^0 = 1$$

Dividing through by $a$ then gives

$$\zbox {a^{-1} = \frac{1}{a}.}$$

Similarly, we obtain

$$\zbox {a^{-M} = \frac{1}{a^M}}$$

for all integer values of $M$, i.e., $\forall M\in{\bf Z}$.