Power is Well-Defined/Rational

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

Let $q$ be a rational number.

Then $x^q$ is well-defined.

Proof
Let $x \in \R_{>0}$ be fixed.

Let $q \in \Q \setminus \set 0$.

Let $\dfrac r s$ and $\dfrac t u$ be two representations of $q$.

That is, $r, s, t, u$ are non-zero integers.

We now show that:
 * $\dfrac r s = \dfrac t u \implies x^{r / s} = x^{t / u}$

So:

Hence the result.