(Please refer to the attached image). When there is a question that has the visual representation for the square root of a number, does it refer to only the principal square root [on GMAT]? So for example, in this question, the numerator would evaluate to only +x (thus in the correct answer, numerator is |x|).

That is correct.

Just for the record, note that "+x" can actually be a negative number (if x itself is negative), although I understand what you're trying to write.

:)

This one is from GMATPrep as well.

(Please see the attached image)
I thought that the answer to this one would be D but surprisingly, it turned out to be A. After looking at the answer, I think it makes sense but I want to make sure that I am following the correct line of reasoning. The solution to this question also hinges on the assumption that on the GMAT, the square root symbol denotes the principal sq root (or the positive square root)

So if I plug in x= -4, what it means is
A=sq root(-(-4)*|-4|)=4. So the answer which is -x means -(-4)=4. Does that sound about right?

Yes. Those are confusing, aren't they? You basically have to hold in mind that saying "x" represents a negative number. If you work through the problem, you end up with the positive version of that number, so to represent that positive version in terms of x, you have to put a negative sign in front - hence, -x.

It is easier if you work through it with a real number, rather than trying to use the variables.

What am I doing wrong?

sqroot(-x*|x|). I evaluate the equation for x and -x.

x: sqroot(-x*x) = sqroot(-x^2) = -x
-x: sqroot(x*-x) = sqroot(x^2) = x

x could be -x or x so therefore |x|.

I am not saying that is answer because the all knowing GMAT says no, but that's what I got and I want to know why it's incorrect.

Thanks,

For the first problem, can't you say (x^2)^(1/2) = x^1 = x so then you have x/x = 1 ?

Your problem is the dropped negative sign in your second evaluation:
sqroot(x*-x) = sqroot(-x^2), NOT sqroot(x^2)

This is also extremely confusing because you then start thinking - wait, how can I solve this, because I've got a negative sign under a square root, which leads to imaginary numbers? (Though, of course, it isn't really, because x is a negative number, so "-x" is a positive number. But you don't know that when you're trying to manipulate this.)

So the real answer to your question "What am I doing wrong" is: you're doing algebra. Do this one with real numbers instead. :)

You can, but you'd be covering only one of two options, the positive root option. Try it with real numbers, one positive and one negative (since this is a question of roots).

SQRT(x^2)/x
If x = 2, then we get SQRT4/2 = 2/2 = 1.
If x = -2, then we get SQRT4/(-2) = 2/(-2) = -1.

Which one's right? They're both possibilities, so we can't choose just one as the right answer.

Instead, answer E (lxl/x) takes care of the problem by accommodating both the positive and negative possibilities.