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If x<0, then root(-x*|x|) is:

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If x<0, then root(-x*|x|) is: [#permalink] New post 29 Jul 2009, 15:42
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If x<0, then \sqrt{-x|x|} is:

A. -x
B. -1
C. 1
D. x
E. \sqrt{x}
[Reveal] Spoiler: OA
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Re: If x<0 [#permalink] New post 16 Sep 2010, 18:14
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saxenashobhit wrote:
hailtothethief23 wrote:
It is actually A.

suppose x is -2 then you have sqrt(2*2) = sqrt(4) = 2 = -x

note that x < 0 as otherwise the function does not exist.

I had same question today

\sqrt{4} = + or - 2. So answer should be + or - x. I don't get how can OA be -x


SOME NOTES:

1. GMAT is dealing only with Real Numbers: Integers, Fractions and Irrational Numbers.

2. Any nonnegative real number has a unique non-negative square root called the principal square root and unless otherwise specified, the square root is generally taken to mean the principal square root.

When the GMAT provides the square root sign for an even root, such as \sqrt{x} or \sqrt[4]{x}, then the only accepted answer is the positive root.

That is, \sqrt{25}=5, NOT +5 or -5. In contrast, the equation x^2=25 has TWO solutions, +5 and -5. Even roots have only non-negative value on the GMAT.

Odd roots will have the same sign as the base of the root. For example, \sqrt[3]{125} =5 and \sqrt[3]{-64} =-4.

3. \sqrt{x^2}=|x|.

The point here is that as square root function can not 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|.


BACK TO THE ORIGINAL QUESTION:

1. If x<0, then \sqrt{-x*|x|} equals:
A. -x
B. -1
C. 1
D. x
E. \sqrt{x}

\sqrt{-x*|x|}=\sqrt{(-x)*(-x)}=\sqrt{x^2}=|x|=-x. Note that as x<0 then -x=-negative=positive, so \sqrt{-x*|x|}=-x=positive as it should be.

Or just substitute the some negative x, let x=-5<0 --> \sqrt{-x*|x|}=\sqrt{-(-5)*|-5|}=\sqrt{25}=5=-(-5)=-x.

Answer: A.

Hope it's clear.
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Re: If x<0 [#permalink] New post 16 Sep 2010, 17:55
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hailtothethief23 wrote:
It is actually A.

suppose x is -2 then you have sqrt(2*2) = sqrt(4) = 2 = -x

note that x < 0 as otherwise the function does not exist.

I had same question today

\sqrt{4} = + or - 2. So answer should be + or - x. I don't get how can OA be -x
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Re: If x<0 [#permalink] New post 03 Dec 2012, 23:46
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nss123 wrote:
I'm having trouble understanding how they get to the answer, though it should be very simple:

If x<0, then \sqrt{-x|x|} is:
A) -x
B) -1
C) 1
D) x
E) \sqrt{x}
I got D, b/c I'm thinking, if x is negative, then taking the opposite of that will be positive, mulitiplied by the absolute value of x, which is also positive. That gives you x^2, which you take the square root of, to get x.

Any help would be great.


Square root of any number CANNOT be negative... The answer is quite apparent and should not take more than 10 seconds to figure out.. B & D can be immediately eliminated.. There is no evidence to show C.(C takes a constant value for the root of a variable) and E is an imaginary number... A is clearly the right answer..
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Re: If x<0 [#permalink] New post 29 Aug 2013, 08:06
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Are you referring to this:
\sqrt{x^2}=x, if x\geq{0};
\sqrt{x^2}=-x, if x<0.



If yes, then you can put = sign in either case.[/quote]


Yes I was asking for mod definition only, so how to decide which way to go , I mean

Defn1 : |X| = +x when x> 0, and |x|=-x when x is less than eq to zero
Defn2: |X| = +x when xis greater than eq 0, and |x|=-x when x is less than zero

Which defn to go for and when, totally confused in this. Please adcvise.
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Re: If x<0 [#permalink] New post 02 Aug 2009, 17:14
It is actually A.

suppose x is -2 then you have sqrt(2*2) = sqrt(4) = 2 = -x

note that x < 0 as otherwise the function does not exist.
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Re: If x<0 [#permalink] New post 02 Aug 2009, 22:55
nss123 wrote:
I'm having trouble understanding how they get to the answer, though it should be very simple:

If x<0, then \sqrt{-x|x|} is:
A) -x
B) -1
C) 1
D) x
E) \sqrt{x}

I got D, b/c I'm thinking, if x is negative, then taking the opposite of that will be positive, mulitiplied by the absolute value of x, which is also positive. That gives you x^2, which you take the square root of, to get x

Any help would be great.


are you sure the question is correct . I remember seeing the question as sqrt(-x/|x|)
In that case ans is -1.
but the problem you have stated, the ans should be x.
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Re: If x<0 [#permalink] New post 16 Sep 2010, 19:14
I got that wrong, thanks "Bunuel"
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Re: If x<0 [#permalink] New post 16 Sep 2010, 20:15
Thanks. Only catch which I wasn't aware was that on square roots I should consider positive value alone. Although this is wrong based on general Math
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Re: If x<0 [#permalink] New post 03 Dec 2012, 19:50
Thank you for this post.
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Last edited by mbaiseasy on 06 Dec 2012, 04:57, edited 1 time in total.
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Re: If x<0, then root(-x*|x|) is: [#permalink] New post 06 Dec 2012, 04:56
Let x=-5
\sqrt{-(-5)|-5|}=\sqrt{25}=5

5 = -(-5) = -x

Answer: -x or A
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Re: If x<0 [#permalink] New post 29 Aug 2013, 03:47
Bunuel wrote:
saxenashobhit wrote:
hailtothethief23 wrote:
It is actually A.

suppose x is -2 then you have sqrt(2*2) = sqrt(4) = 2 = -x

note that x < 0 as otherwise the function does not exist.

I had same question today

\sqrt{4} = + or - 2. So answer should be + or - x. I don't get how can OA be -x


SOME NOTES:

1. GMAT is dealing only with Real Numbers: Integers, Fractions and Irrational Numbers.

2. Any nonnegative real number has a unique non-negative square root called the principal square root and unless otherwise specified, the square root is generally taken to mean the principal square root.

When the GMAT provides the square root sign for an even root, such as \sqrt{x} or \sqrt[4]{x}, then the only accepted answer is the positive root.

That is, \sqrt{25}=5, NOT +5 or -5. In contrast, the equation x^2=25 has TWO solutions, +5 and -5. Even roots have only non-negative value on the GMAT.

Odd roots will have the same sign as the base of the root. For example, \sqrt[3]{125} =5 and \sqrt[3]{-64} =-4.

3. \sqrt{x^2}=|x|.

The point here is that as square root function can not 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|.


BACK TO THE ORIGINAL QUESTION:

1. If x<0, then \sqrt{-x*|x|} equals:
A. -x
B. -1
C. 1
D. x
E. \sqrt{x}

\sqrt{-x*|x|}=\sqrt{(-x)*(-x)}=\sqrt{x^2}=|x|=-x. Note that as x<0 then -x=-negative=positive, so \sqrt{-x*|x|}=-x=positive as it should be.

Or just substitute the some negative x, let x=-5<0 --> \sqrt{-x*|x|}=\sqrt{-(-5)*|-5|}=\sqrt{25}=5=-(-5)=-x.

Answer: A.

Hope it's clear.

When x is greater than equal to zero , x is positive

when x is less than zero, x is negative. can we also say as below ?

when x is greater than zero , x is positive,
and when x is less than equal to zero, x is negative .
The main point is can we switch equal condition on either side?
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Re: If x<0 [#permalink] New post 29 Aug 2013, 03:55
Expert's post
ygdrasil24 wrote:
Bunuel wrote:
saxenashobhit wrote:
I had same question today

\sqrt{4} = + or - 2. So answer should be + or - x. I don't get how can OA be -x


SOME NOTES:

1. GMAT is dealing only with Real Numbers: Integers, Fractions and Irrational Numbers.

2. Any nonnegative real number has a unique non-negative square root called the principal square root and unless otherwise specified, the square root is generally taken to mean the principal square root.

When the GMAT provides the square root sign for an even root, such as \sqrt{x} or \sqrt[4]{x}, then the only accepted answer is the positive root.

That is, \sqrt{25}=5, NOT +5 or -5. In contrast, the equation x^2=25 has TWO solutions, +5 and -5. Even roots have only non-negative value on the GMAT.

Odd roots will have the same sign as the base of the root. For example, \sqrt[3]{125} =5 and \sqrt[3]{-64} =-4.

3. \sqrt{x^2}=|x|.

The point here is that as square root function can not 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|.


BACK TO THE ORIGINAL QUESTION:

1. If x<0, then \sqrt{-x*|x|} equals:
A. -x
B. -1
C. 1
D. x
E. \sqrt{x}

\sqrt{-x*|x|}=\sqrt{(-x)*(-x)}=\sqrt{x^2}=|x|=-x. Note that as x<0 then -x=-negative=positive, so \sqrt{-x*|x|}=-x=positive as it should be.

Or just substitute the some negative x, let x=-5<0 --> \sqrt{-x*|x|}=\sqrt{-(-5)*|-5|}=\sqrt{25}=5=-(-5)=-x.

Answer: A.

Hope it's clear.

When x is greater than equal to zero , x is positive

when x is less than zero, x is negative. can we also say as below ?

when x is greater than zero , x is positive,
and when x is less than equal to zero, x is negative .
The main point is can we switch equal condition on either side?



Are you referring to this:
\sqrt{x^2}=x, if x\geq{0};
\sqrt{x^2}=-x, if x<0.

If yes, then you can put = sign in either case.
_________________

NEW TO MATH FORUM? PLEASE READ THIS: ALL YOU NEED FOR QUANT!!!

PLEASE READ AND FOLLOW: 11 Rules for Posting!!!

RESOURCES: [GMAT MATH BOOK]; 1. Triangles; 2. Polygons; 3. Coordinate Geometry; 4. Factorials; 5. Circles; 6. Number Theory; 7. Remainders; 8. Overlapping Sets; 9. PDF of Math Book; 10. Remainders; 11. GMAT Prep Software Analysis NEW!!!; 12. SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) NEW!!!; 12. Tricky questions from previous years. NEW!!!;

COLLECTION OF QUESTIONS:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS ; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


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Re: If x<0 [#permalink] New post 29 Aug 2013, 08:51
Expert's post
ygdrasil24 wrote:
Are you referring to this:
\sqrt{x^2}=x, if x\geq{0};
\sqrt{x^2}=-x, if x<0.



If yes, then you can put = sign in either case.



Yes I was asking for mod definition only, so how to decide which way to go , I mean

Defn1 : |X| = +x when x> 0, and |x|=-x when x is less than eq to zero
Defn2: |X| = +x when xis greater than eq 0, and |x|=-x when x is less than zero

Which defn to go for and when, totally confused in this. Please adcvise.[/quote]

It does NOT matter in which part you put = sign.
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NEW TO MATH FORUM? PLEASE READ THIS: ALL YOU NEED FOR QUANT!!!

PLEASE READ AND FOLLOW: 11 Rules for Posting!!!

RESOURCES: [GMAT MATH BOOK]; 1. Triangles; 2. Polygons; 3. Coordinate Geometry; 4. Factorials; 5. Circles; 6. Number Theory; 7. Remainders; 8. Overlapping Sets; 9. PDF of Math Book; 10. Remainders; 11. GMAT Prep Software Analysis NEW!!!; 12. SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) NEW!!!; 12. Tricky questions from previous years. NEW!!!;

COLLECTION OF QUESTIONS:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS ; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


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Re: If x<0 [#permalink] New post 29 Aug 2013, 10:19
It does NOT matter in which part you put = sign

Ok. Thanks. I ll check.
Re: If x<0   [#permalink] 29 Aug 2013, 10:19
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