# roots written as exponents

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.

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$a^{\frac{m}{n}}$ is just ${}^n\sqrt{a^m}$. –  Kevin Carlson Sep 26 '12 at 13:38
Take $m=1=n$ and $a=-1$. $(-1)^{\frac{2}{2}} = (-1)^1 = -1$ whereas $\sqrt{(-1)^{2}} = \sqrt{1} = 1$ –  xavierm02 Sep 26 '12 at 13:44
@KevinCarlson: You can typeset a non-square root with \sqrt[n]{a^m}. –  Henning Makholm Sep 26 '12 at 14:33
Thanks, @HenningMakholm. –  Kevin Carlson Sep 26 '12 at 14:37

$\frac{a^{1/3}}{a}=a^{1/3-1}=a^{-2/3}=(a^{2/3})^{-1}=\frac{1}{a^{2/3}}$

Hope this helps!

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makes perfect sense. though still ~ is there a way to think of it as a root? –  foaly Sep 26 '12 at 13:38
Exactly as Kevin Carlson says :) –  Fernando Domene Sep 26 '12 at 13:46
In general, you have $x^{\frac{a}{b}}=(\sqrt[b]{x})^{a}$ for positive integers $a,b\in\mathbb{N}$ and $x\geq0$.
i wasn't (even) aware of the fact that $\sqrt[b]{x^a} = \sqrt[b]{x}^a$... jeez i really got a problem with my basics x.x –  foaly Sep 26 '12 at 13:52