Power is Well-Defined/Rational

Theorem
Let $x$ be a positive  real number.

Let $q$ be a rational number.

Then $x^q$ is well-defined.

Proof
Fix $x \in \R_{> 0}$.

Let $q \in \Q \setminus \left \{ 0 \right \}$.

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

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

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

Hence the result.