Am I missing something? Do I need to specify a condition like x belongs to R for it to work?
Indeed. By default all variables are assumed Complex unless told otherwise
FullSimplify[Sqrt[x^2], x \[Element] Reals]
This is how you specify that the variable is real. Note that the result gives back:
Abs[x]
rather than x, because the square root does not return negative values. Only if one assumed positive numbers then it returns x:
FullSimplify[Sqrt[x^2], x \[Element] Reals \[And] x > 0]
gives back:
x
