Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized for You

we will pick new questions that match your level based on your Timer History

Track Your Progress

every week, we’ll send you an estimated GMAT score based on your performance

Practice Pays

we will pick new questions that match your level based on your Timer History

Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.

It appears that you are browsing the GMAT Club forum unregistered!

Signing up is free, quick, and confidential.
Join other 500,000 members and get the full benefits of GMAT Club

Registration gives you:

Tests

Take 11 tests and quizzes from GMAT Club and leading GMAT prep companies such as Manhattan GMAT,
Knewton, and others. All are free for GMAT Club members.

Applicant Stats

View detailed applicant stats such as GPA, GMAT score, work experience, location, application
status, and more

Books/Downloads

Download thousands of study notes,
question collections, GMAT Club’s
Grammar and Math books.
All are free!

Thank you for using the timer!
We noticed you are actually not timing your practice. Click the START button first next time you use the timer.
There are many benefits to timing your practice, including:

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 \(\sqrt{x^2}=|x|\).
_________________

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

Show Tags

06 Sep 2017, 02:25

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 \(\sqrt{x^2}=|x|\).

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

Show Tags

07 Sep 2017, 16:48

2

This post received KUDOS

1

This post was BOOKMARKED

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 \(x<0\)...

sinhap07How is mod of -x equal to -x? Shouldnt it be x?

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.

\(-xy\) 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 \(x<0\)...

sinhap07How is mod of -x equal to -x? Shouldnt it be x?

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.

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.