TITLE: Why is ${x^{\frac{1}{2}}}$ the same as $\sqrt x $?
QUESTION [37 upvotes]: Why is ${x^{\frac{1}{2}}}$ the same as $\sqrt x $?

I'm currently studying indices/exponents, and this is a law that I was told to accept without much proof or explanation, could someone explain the reasoning behind this.
Thank you.

REPLY [6 votes]: For $x\geq 0 $ and by definition $\sqrt x$ is the positive real $y$ such that $y^2=x$
and since 
$$\left(x^p\right)^q=x^{pq}$$
then for $p=\frac1 2$ and $q=2$ we see that $x^{1/2}=\sqrt x$