
Squaring a square root
I realize that a variable expression or just a variable that is squared and square rooted cannot be simplified to the expression itself (due to negation of negatives after squaring).
$\displaystyle \sqrt{x^2}\neq x $ for all $\displaystyle x<0 $
I logically thought about this and came to the conclusion that this the square of a square root of an expression must be the absolute value of the expression. Is this true?
$\displaystyle \sqrt{x^2} = x $ for all $\displaystyle x $ ??

Indeed. $\displaystyle \sqrt{x^2} = x$ for any $\displaystyle x \in \mathbb{R}$ :)

Thanks, Ray. I had only conjectured this, but didn't know if this was true.