It appears that i'm not quite sure anymore how to write roots as exponents, and how to work with them. I know $\sqrt[3]{a}$ is written $a^{\frac{1}{3}}$, but I don't know how to handle them in things like fractures ($\frac{a^{\frac{1}{3}}}{a}$ appears to be $\frac{1}{a^\frac{2}{3}}$), or what to think of when seeing something like $a^\frac{2}{3}$
It would be nice if you could give me a brief, but as complete as possible description on this :)
thank you.