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 $ ??