I saw the following equation : $$x^{2/3}$$
Instead of the correct expansion $$x^{2/3} = ( \sqrt[3]{x} ) ^ 2$$
I made the following mistake $$x^{2/3} = ( \sqrt[3]{x^2} ) $$
In order not to make the same mistake again I tried to formulate my mistake in terms of some rule but I do not really see what rule I trespassed. I would appreciate if could let me know.
Edit : I see that in this particular case $$( \sqrt[3]{x} ) ^ 2 = ( \sqrt[3]{x^2} ) $$
but what if the equation was $$x^{2/4}$$
then
$$( \sqrt[4]{x} ) ^ 2 \neq ( \sqrt[4]{x^2} ) $$
as two functions would have different domains. I know, I could simplify the 2/4 to 1/2 but for the sake of argument I choose not to and I am not aware of any rule that forces me to do the simplification.
Instead, try to prove that indeed $$\sqrt[3]{x^2} \ =\ (\sqrt[3]x)^2$$ for all real $x$.