Why is $|x|$ defined as $\sqrt{x^2}$ instead of $(\sqrt{x})^2$? I can't seem to understand this even though it might be utterly simple for some people. For me, saying $|x|=\sqrt{x^2}$ is a bit weird since $\sqrt{x^2}$ doesn't force positivity as there are always two possible square roots of a number, 
 $\sqrt{x}=+\sqrt x, -\sqrt{x}$. Then why is it still used everywhere? Is there another way of interpreting this?
I think $|x|=(\sqrt x)^2$ does the job much better.

First of all, $\sqrt{x^2}$ is necessarily positive.  There is an implicit + sign.
Secondly, the notation is meant to represent the magnitude of a given vector.  Sure, your definition works in the case of a scalar.  But for vector magnitudes(with 2 or more components), your definition breaks down(i. e. $|\binom{x}{y}| = \sqrt{x^2+y^2} \ne (\sqrt{x+y})^2$).
