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 \(x\).
$x^n$ 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 \(n\):
Proofs: 1
Corollaries: 1
Definitions: 2 3
Proofs: 4 5 6 7 8
Propositions: 9 10 11 12 13 14 15 16
Theorems: 17
