What is (x^2y^2)^1/2 if x 0 ?

Question

What is  \sqrt{x^2*y^2}  if x < 0 and y > 0?

(A) –xy
(B) xy
(C) –|xy|
(D) |y|x
(E) No solution

KarishmaB · Accepted Answer

Note that  \sqrt{x^2} = |x| . It is not x, it is |x|. When we talk about square root, it implies the principal square root i.e. just the positive square root.

\sqrt{x^2*y^2} = |x|*|y|

Now, if x < 0,  |x| = -x 
If y > 0,  |y| = y

Hence,  |x|*|y| = -x*y

Answer (A)

Bunuel · Answer

If you plug x=-2 and y=2, then  \sqrt{x^2*y^2}=4 :

(A) –xy = 4. Match.
(B) xy = -4. Discard.
(C) –|xy| = -4. Discard.
(D) |y|x = -4. Discard.

So, plunging in gives the same answer.

One could discard B, C and D right away. The square root function cannot give negative result, while all these options are negative, which means that they cannot be correct answers:

(A) –xy = -(negative)(positive) = positive. OK.
(B) xy = (negative)(positive) =  negative . Discard.
(C) –|xy| = -positive =  negative . Discard.
(D) |y|x = positive*negative =  negative . Discard.

Hope it's clear.

PareshGmat · Answer

It seems there is a risk if we plugin numbers eg: -2 & 2 which may not give an undisputed answer as computed by Karishma above

Experts, have your say..

simba19 · Answer

Karishma--

I understand the algebraic method of solving the problem, but I don't understand why plugging numbers isn't accurate on these kinds of problems. Could you explain? Thanks!

KarishmaB · Answer

As explained by Bunuel above, plugging-in numbers isn't necessarily a problem. Sometimes, it can be problematic because multiple options may match. Then you would need to try another set of numbers on the options that matched. Usually, you will get one correct answer within two rounds.

GMatAspirerCA · Answer

Hi Bunuel and Karisma, 
does that mean square root of 4 is always 2 and  not -2 ? 
I thought it had two options just like x to the power 2 = 4 gives, x = 2 and x = -2.

Bunuel · Answer

When the GMAT provides  the square root sign  for an even root, such as  \sqrt{x}  or  \sqrt {x} , then the  only accepted answer is the positive root . That is,  \sqrt{4}=2 , NOT +2 or -2.

In contrast, the equation  x^2=4  has TWO solutions, +2 and -2.

Hope this helps.

ameyaprabhu · Answer

What's the difference between A and C?

Both will give us the same answer right?

A = - (of a positive xy)
B = - (of a positive xy)

KarishmaB · Answer

You are given that x < 0 (say x is -2) and y > 0 (say y is 5). 
So xy will be -2*5 = -10.

Hence xy will actually be a negative number.

Option (A) = - xy = - (-10) = 10 (a positive number)

Option (C) = - |xy| = - |-10| = -10 (a negative number)

LakerFan24 · Answer

VeritasPrepKarishma

am i correct in saying that  \sqrt{x^{2}}  is NOT the same as writing  \sqrt{(x^{2})} , so is this the reason here that  \sqrt{x^{2}}  = (-x)?

is this what the problem is testing? because otherwise, i cannot understand how an absolute value is negative -- an absolute value can never be negative, because it is the measure of how far away numbers are on a number line

Bunuel · Answer

\sqrt{x^{2}}  is the same as  \sqrt{(x^{2})} .

\sqrt{x^{2}}=|x| . We are given that x is negative. When x < 0, we know that |x| = -x, so  \sqrt{x^{2}}=|x|=-x . Notice that since x is negative, -x = -negative = positive, thus both the square root and the absolute value return positive result.

KarishmaB · Answer

In addition to what Bunuel said above, let me also add a line on how I explain this in words: You are right. An absolute value can never be negative. So |x| will never be negative. But it can be equal to -x. When? When x itself is negative. Note that a variable x CAN stand for a negative value too. So if x itself is negative, -x becomes POSITIVE. And that is the case in which |x| is equal to -x (which is positive).

Krishnahelps · Answer

Hi  KarishmaB   Bunuel , can't be write (x^2y^2)^1/2 =|x*y|. Why do we have to put x and y in separate modulus?

Bunuel · Answer

Both are the same: |xy| = |x|*|y|.

A. · Answer

This question is an example how even simple questions can go wrong if the basics are not clear

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

What is (x^2y^2)^1/2 if x < 0 and y > 0 ?

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions