A real power function $f:\,\mathbb{R}_+\to\mathbb{R}$ , has the form $$f(x) = x^a$$ where $a$ is a given real number.

The power functions comprise the natural power functions $x\mapsto x^n$ , with $n = 0,\,1,\,2,\,\ldots$ the root functions $x\mapsto \sqrt[n]{x} = x^{\frac{1}{n}}$ , with $n = 1,\,2,\,3,\,\ldots$ , and other fraction power functions $x\mapsto x^a$ , with $a$ any fractional number.

Note. The power $x^a$ may of course be meaningful also for other than positive values of $x$ if $a$ is an integer. On the other hand, e.g. $(-1)^{\sqrt{2}}$ has no real values -- see the general power.