Hi,
When you solve an equation you can sometimes say the solution is plus or minus x.
But when you take the square root of a square you have to say that it is the abs value, because then it's a different story as the operations are simply sequential.
Lets see now, if we have (x-a)^2 and we take the square root, it seems like we would get x-a back. But watch what happens...
Lets say a is 3 and x is 5.
(5-3)^2=2^2=4, and sqrt(4)=2, no problem.
Now lets say a is 3 and x is 1.
(1-3)^2=(-2)^2=4, and sqrt(4)=2, same as before.
So simply put, in both cases we got abs(x-a), not plus or minus (x-a). So the answer here is not plus or minus (x-a) it's abs(x-a).
So the inverse operation (square root is the inverse operation of the squaring function) does not always yield the original expression. Something gets lost in the translation. So it doesnt matter what x or a is, we always get abs(x-a) and that's it.
I think if we do (x-a)^3 and then cube root we get the same thing back, but that's different because we dont loose the sign. We do have to be careful though.