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.

You can see that $\sqrt{x^2}$ is defined for all $x\in\Bbb R$, while $(\sqrt{x})^2$ is defined only for $x\ge 0$. And that's a huge difference.
Furthermore, $|x|$ is not defined as $(\sqrt x)^2$. Instead, $|x|$ is defined as $x$ if $x\ge 0$ and as $-x$ if $x<0$.
