I would like to have a bit more specific details on what makes a problem like this

\sqrt{b^1^8}=\left| b^9 \right|

From your second post it looks like you mean this:

End up with an absolute value. I am having trouble comprehending what determines the answer to be in a form of an absolute value.

Whatever b is (+ or -) b^(18) will be positive (or 0 if b = 0). But b^9 will be negative if b is negative. So they can't be equal unless they come up with the same answers. The sqrt will always give a positive; the 9th power won't. So you have to put the positive-only back in by doing the bars.