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Positive Integer Exponents

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


$\displaystyle \zbox {a^n \isdef \underbrace{a\, a \, a \,\cdots \,a \, a}_{\mbox{$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.
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Euler's Identity