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Intern  Joined: 16 Nov 2013
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Location: United States
Concentration: Entrepreneurship, General Management
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What is root(x^2•y^2) if x < 0 and y > 0?  [#permalink]

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Difficulty:   25% (medium)

Question Stats: 66% (00:52) correct 34% (01:06) wrong based on 1467 sessions

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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
Math Expert V
Joined: 02 Sep 2009
Posts: 58381
Re: What is root(x^2•y^2) if x < 0 and y > 0?  [#permalink]

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11
19
Nai222 wrote:
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.

$$-xy$$ 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 $$\sqrt{x^2}=|x|$$.
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Math Expert V
Joined: 02 Sep 2009
Posts: 58381
Re: What is root(x^2•y^2) if x < 0 and y > 0?  [#permalink]

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11
registerincog wrote:
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

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

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Re: What is root(x^2•y^2) if x < 0 and y > 0?  [#permalink]

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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.
Manager  Joined: 29 Sep 2013
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Re: What is root(x^2•y^2) if x < 0 and y > 0?  [#permalink]

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WOW!!! Thank you for that explanation. I definitely get it now.        EMPOWERgmat Instructor V
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GMAT 1: 800 Q51 V49 GRE 1: Q170 V170 What is root(x^2•y^2) if x < 0 and y > 0?  [#permalink]

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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

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Re: What is root(x^2•y^2) if x < 0 and y > 0?  [#permalink]

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for |xy| we have 4 options

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

A
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GRE 1: Q169 V154 Re: What is root(x^2•y^2) if x < 0 and y > 0?  [#permalink]

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We must remember here that x>0 => √x^2 = x else if x is negative => -x
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Re: What is root(x^2•y^2) if x < 0 and y > 0?  [#permalink]

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Bunuel wrote:
Nai222 wrote:
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.

$$-xy$$ 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 $$\sqrt{x^2}=|x|$$.

How is mod of -x equal to -x? Shouldnt it be x?
Senior SC Moderator V
Joined: 22 May 2016
Posts: 3544
What is root(x^2•y^2) if x < 0 and y > 0?  [#permalink]

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3
1
sinhap07 wrote:
Bunuel wrote:
Nai222 wrote:
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.

$$-xy$$ is not negative it's positive. $$xy$$ is negative, thus $$-xy=-(negative)=positive$$.

THEORY:
. . .
What function does exactly the same thing [as the square root sign]? The absolute value function: $$|x|=x$$, if $$x\geq{0}$$ and $$|x|=-x$$, if $$x<0$$...

sinhap07 How is mod of -x equal to -x? Shouldnt it be x?

sinhap07 , I assume you refer to
Quote:
$$|x|= -x$$ , if $$x<0$$

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 [+3 = -(-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.
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Joined: 02 Sep 2009
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What is root(x^2•y^2) if x < 0 and y > 0?  [#permalink]

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3
genxer123 wrote:
sinhap07 wrote:
Bunuel wrote:
$$-xy$$ is not negative it's positive. $$xy$$ is negative, thus $$-xy=-(negative)=positive$$.

THEORY:
. . .
What function does exactly the same thing [as the square root sign]? The absolute value function: $$|x|=x$$, if $$x\geq{0}$$ and $$|x|=-x$$, if $$x<0$$...

sinhap07 How is mod of -x equal to -x? Shouldnt it be x?

sinhap07 , I assume you refer to
Quote:
$$|x|= -x$$ , if $$x<0$$

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

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

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

Hope that helps.

|-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.
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Re: What is root(x^2•y^2) if x < 0 and y > 0?  [#permalink]

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registerincog wrote:
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

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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Re: What is root(x^2•y^2) if x < 0 and y > 0?  [#permalink]

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