Definition:Power (Algebra)/Real Number/Definition 2

Definition
Let $x \in \R_{>0}$ be a (strictly) positive real number.

Let $r \in \R$ be a real number.

Let $f : \Q \to \R$ be the real-valued function defined as:
 * $f \left({ q }\right) = x^q$

where $a^q$ denotes $a$ to the power of $q$.

Then we define $x^r$ as the unique continuous extension of $f$ to $\R$.

Also see
Exponential with Rational Power Permits Unique Continuous Extension, where the existence and uniqueness of such an extension is proven.