It's not a simplification, it's wrong. Sqrt(square(x)) equals abs(x).
replyIt also equals x with appropriate assumptions (x > 0).
replyWell, then sin(x) = x if x is infinitely small
replyso there's an unconditionally correct answer (it's also equal to abs(x) for x>0), and then there is an answer that is only correct for half the domain, which requires an additional assumption.
replysqrt(square(i)) != abs(i)
replySo no, it’s not unconditionally correct either.
Not in general. As people have pointed out elsewhere, it's true if x is real. That isn't always a helpful assumption. (When x is real you can plug that assumption into Mathematica. Then Mathematica should agree with you.)
replyBut consider sqrt(i) = sqrt(exp(i\pi/2)). That's exp(i\pi/4). Your rule would give 1 as the answer. It's not helpful for a serious math system to give that answer to this problem.
When I square 1 I don't get i.