Positive Integer Exponents

The ``original'' definition of exponents which ``actually makes sense'' applies only to positive integer exponents:

$$\zbox {a^n \isdef \underbrace{a\, a \, a \,\cdots \,a \, a}_{\mbox{$n$\ times}}}$$

where $a>0$ is real.

Generalizing this definition involves first noting its abstract mathematical properties, and then making sure these properties are preserved in the generalization.