What is root(x^2•y^2) if x 0?

Question

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

A) -xy
B) xu
C) -|xy|
D) |y|x
E) No solution

Bunuel · Accepted Answer

-xy is not negative it's positive. xy is negative, thus -xy=-(negative)=positive . THEORY: \sqrt{x^2}=|x| . The point here is that as square root function cannot give negative result then \sqrt{some \ expression}\geq{0} . So \sqrt{x^2}\geq{0} . But what does \sqrt{x^2} equal to? Let's consider following examples: If x=5 --> \sqrt{x^2}=\sqrt{25}=5=x=positive ; If x=-5 --> \sqrt{x^2}=\sqrt{25}=5=-x=positive . So we got that: \sqrt{x^2}=x , if x\geq{0} ; \sqrt{x^2}=-x , if x<0 . What function does exactly the same thing? The absolute value function: |x|=x , if x\geq{0} and |x|=-x , if x<0 . That is why \sqrt{x^2}=|x| .

Bunuel · Answer

\sqrt{x^2 y^2}=\sqrt{(xy)^2} = |xy| . Since x is negative and y is positive, then xy is negative, thus  |xy|=-xy .

Answer: A.

Nai222 · Answer

hey,

Can you please explain why it is negative? I thought when the two number are enclosed within an absolute value bracket, it becomes positive.

Nai222 · Answer

WOW!!! Thank you for that explanation. I definitely get it now. :-D

EMPOWERgmatRichC · Answer

Hi Nai222,

Since your post was from over a year ago, you might not still be around to read this. This question can be solved by TESTing VALUES:

We're told X < 0 and Y > 0

Let's TEST:
X = -2
Y = +2

So, we'd have \sqrt{(4)(4)} = +4

Using those values in the answers, we get...

Answer A: -(-2)(2) = +4 This IS a match

Answer B: (-2)(2) = -4 NOT a match

Answer C: - |(-2)(2)| = -4 NOT a match

Answer D: |2|(-2) = -4 NOT a match

Answer E: No solution. NOT a match

Final Answer:  B

GMAT assassins aren't born, they're made,
Rich

Temurkhon · Answer

for |xy| we have 4 options

xy, x-y,  -xy , -x-y

A

stonecold · Answer

We must remember here that x>0 => √x^2 = x else if x is negative => -x

sinhap07 · Answer

How is mod of -x equal to -x? Shouldnt it be x?

generis · Answer

sinhap07  , I assume you refer to
 
For me, the rule can be counterintuitive. I've seen a few ways to think about the rule. Maybe they will help.

If x is negative , then |x| = -x

Possible ways to think about the rule:

1. "The negative of a negative is positive."

Think of the negative sign as signifying "opposite."

That is, the negative sign functions as "the negative of a negative number." And the negative of a negative number is positive. See  Bunuel  above in bold.

Let x = -3. Per |x| = -x:

|-3| = 3, and +3 is the opposite of -3, thus +3 = -(-3)

2. OR think: "The negative sign on RHS means (-1) multiplied by a negative number  x ," thus

|x| = -x 
|x| = (-1)(x) 
|x| = (-1)(negative #)
|x| = a positive number

3. OR think (similar to #1): "in this rule there is a hidden minus sign."

With a number, the "two negatives" are easy to see

|-3| = 3
|-3| =  -(-3)

BUT: |x| = -(x) = -x

With variable  x , it is easy to forget that there ARE two negative signs.

With the variable, there is only one minus sign on RHS... because the negative variable  x  already "contains" a minus sign.

We just don't (can't) write the minus sign twice with the variable.

|-3| = -(-3) = 3
|x| = -(x) = -x

Those two equations are functionally equivalent.

4. Summary - use any negative number, substituted for x, to see that, if x < 0 , then |x| = -x. Reasons:

|-3| = 3, where +3 is the opposite of -3  ; RHS is the negative of a negative number

|-3| = 3 = (-1)(-3)

|-3| = -(-3) = 3

The absolute value IS positive (or nonnegative). The sign of a negative variable can obscure that fact.

Hope that helps.

Bunuel · Answer

To add:

|-x| = |x|. One way to think about it is that |-x| is the distance between -x and 0 on the number line. Similarly, |x| is the distance between x and 0 on the number line. Obviously -x and x are the same distance from 0. For example, -3 and 3 are the same distance from 0; 2 and -2, are the same distance from 0...

Next, when x is 0 or negative, the rule says that |x| = -x. The absolute value cannot be negative and this rule is not violated here. For example, say x = -10, then |-10| = -(-10) = 10 = positive or generally when x is negative |x| = -x = -negative = positive. Or using the distance concept again |-10| is the distance from -10 to 0, which is 10.

Hope it helps.

CyberStein · Answer

If x<0, then x = (-)
if y > 0, then y = (+)
root x^2y^2 = (+), for, given the exponent is even, the power of (-) is (+)

B) xy = (-)(+) = (-) therefore insufficient 
C) (-)(+) = (-)
D)(+)(-) = (-)

A) (-)(-) = (+)

Therefore, the answer is A-xy

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What is root(x^2•y^2) if x < 0 and y > 0?

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