Root is Commutative

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

Let $a$ and $b$ be nonzero integers.

Then $\sqrt[a]{\sqrt[b]x} = \sqrt[b]{\sqrt[a]x}$.

Proof
Let $y = \sqrt[a]{\sqrt[b]x}$. Then