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

1.Proof: (related to "Nth Powers") 

2.Proposition: Derivative of the n-th Power Function