What is wrong with this proof that x = |x|

Question

First, take $x$. $x=x^1$. $1=\frac22$, so $x^1 = x^\frac22 = (\sqrt{x})^2 = |x|$. Therefore $x = |x|$.

I checked my proof and think that all my other steps are valid. I think my mistake has to do with $x^\frac22 = (\sqrt{x})^2$

So, if there is anything wrong with this step, what is it?

Answer 1

You must use the definition of $|x|$ and consider the cases $x > 0$ and $x \leq 0$.

What you have done does not work as $\sqrt{x}$ is not defined for $x < 0$ when working over the real numbers. Even then, after you square the square root of $x$ you are left with $|x| = |x|$...