(sqrt(x^2))^2 Absolute value question

[mt057]

When you perform the operation (sqrt(x^2))^2, you get the absolute value of x^2? I thought that when you square a square root you get the number.

I just need some help wrapping my head around it. I have not taken Algebra in a long time and it seems to be one of my weak points. I got a problem wrong in class and did not have time to have my instructor explain it to me. I will have to catch him in his office hours next week and I am very curious so I do not make the mistake again.

 

[pka]

When you perform the operation (sqrt(x^2))^2, you get the absolute value of x^2? I thought that when you square a square root you get the number.

You have been mislead on this point. Here is why. \(\displaystyle \large |a^2|=|a|^2=a^2\).

So \(\displaystyle \large |(\sqrt{x^2})|^2=(|x|^2)=x^2\)

 

[Ishuda]

In context, your statement is correct and you did get both the number itself, x², and the absolute value of x² since
\(\displaystyle | x |^2 =| x^2 | =x^2\).
Notice that the second part of that equation is not true of all exponents since, for example,
\(\displaystyle | -1 |^3 =| (-1)^3 | = 1 \ne (-1)^3 = -1\).