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## Proposition: Nth Powers

Let $$n\ge 0$$ be an integer. The exponentiation of a real number $$x\in\mathbb R$$ defines a function $$x^n:\mathbb R\to\mathbb R$$,
$x^n:=\cases{1&\text{if }n=0\\ \underbrace{x\cdot\ldots\cdot x}_{n\text{ times}}&\text{if }n > 0,\\ \underbrace{x^{-1}\cdot\ldots\cdot x^{-1}}_{-n\text{ times}}&\text{if }n < 0, }$
called the $$n$$-th power of the number $$x$$.

### Note

$x^n$ can also be written as the generalized power of $x$, i.e. as $$x^n=\exp_x\left(n\right).$$

The following interactive picture demonstrates the exponentiation for different values of the exponent $$n$$:

| | | | | created: 2015-08-27 20:17:01 | modified: 2020-03-29 17:30:19 | by: bookofproofs