Question

I have a question regarding absolute value:

|x| = \sqrt{(x)^2}
If this is the case, then why doesn't the square and the square root just cancel each other out, leaving you with:

|x| = x
?

Answer 1

Put in x = -1 and you will see why:

\[x^{2} = (-1)^2 = 1 \ \hbox{hence} \ \sqrt{x^2} = \sqrt{1} = 1.\]

Answer 2

I don't understand.

Answer 3

Your argument is that |x| = x for every real number x. Using the example of x = -1 you can see that this is in fact not the case, as
\[|x| = |-1| = 1\]
but x = -1.

Hence it is not the case for x = -1 that |x| = x. And therefore it is not the case in general that |x| = x for all x.

Answer 4

\[\sqrt{x^2}=x if x>0\]
\[\sqrt{x^2}=-x if x<0\]
\[\sqrt{x^2}=0 if x=0\]