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

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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