While going over how to simplify expressions with radicals with my precalculus class, I ran into the problem of how to explain that $\sqrt[4]{9}$ can be simplified to $\sqrt{3}$. The best I could come up with was along the lines of you "just have to see that the fourth-root can be broken up into two square roots". So something like: $$ \sqrt[4]{9} = \sqrt[2]{\sqrt[2]{9}} = \sqrt[2]{3} \quad\quad\text{or}\quad\quad \sqrt[4]{9} = 9^{\frac{1}{4}} = (9^{\frac{1}{2}})^{\frac{1}{2}} = (3)^{\frac{1}{2}} = \sqrt{3} $$
The step of breaking up the one-fourth seems kinda tricky, and like something students wouldn't notice if they are in the mindset to simplify an expression. Is there a better way to explain this so that students will more immediately see that $\sqrt[4]{9}$ can be simplified? I am looking for an explanation that is more intuitive than just stating the rule, "If the power of any divisors of the radicand divides the index of the radical, then ...", because it seems like there should be some simpler explanation.
It might be better to say $$ \sqrt[4]{9} = 9^{\frac{1}{4}} = (3^{2})^{\frac{1}{4}} = (3)^{\frac{2}{4}} = (3)^{\frac{1}{2}} = \sqrt{3} $$ but turning the $9$ into $3^2$ will still run counter to the students' mindset of simplify.