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Re: Square root and Modulus [#permalink]
02 Sep 2010, 13:07
1
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Expert's post
mbafall2011 wrote:
udaymathapati wrote:
If x < 0, then \sqrt{-x} •|x|) is A. -x B. -1 C. 1 D. x E. \sqrt{x}
what is the source of this question. I havent seen any gmat question testing imaginary numbers
GMAT is dealing only with Real Numbers: Integers, Fractions and Irrational Numbers. So you won't see any question involving imaginary numbers.
This question also does not involve imaginary numbers as expression under the square root is non-negative (actually it's positive): we have \(\sqrt{-x*|x|}\) --> as \(x<0\) then \(-x=positive\) and \(|x|=positive\), so \(\sqrt{-x*|x|}=\sqrt{positive*positive}=\sqrt{positive}\).
Re: Square root and Modulus [#permalink]
08 Dec 2010, 11:34
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shrive555 wrote:
Bunel: i've already seen all the explanation just one more question.
as If \(x<0\), then \(\sqrt{-x*|x|}\) Ans is -x is Answer of the question depends on the condition x<0 or it depends on the sqrt (even root)
lets keep the condition same i.e x<0 and take odd root say cube root. i.e \(\sqrt[3]{-x*|x}|\) . what would be the answer, would it be x then ?
Thanks
About the odd roots: odd roots will have the same sign as the base of the root. For example, \(\sqrt[3]{125} =5\) and \(\sqrt[3]{-64} =-4\).
So, if given that \(x<0\) then \(|x|=-x\) and \(-x*|x|=(-x)*(-x)=positive*positive=x^2\), thus odd root from positive \(x^2\) will be positive.
But \(\sqrt[3]{-x*|x}|\) will equal neither to x nor to -x: \(\sqrt[3]{-x*|x|}=\sqrt[3]{x^2}=x^{\frac{2}{3}}\), for example if \(x=-8<0\) then \(\sqrt[3]{-x*|x|}=\sqrt[3]{x^2}=\sqrt[3]{64}=4\).
If it were \(x<0\) and \(\sqrt[3]{-x^2*|x}|=?\), then \(\sqrt[3]{-x^2*|x}|=\sqrt[3]{-x^2*(-x)}=\sqrt[3]{x^3}=x<0\). Or substitute the value let \(x=-5<0\) --> \(\sqrt[3]{-x^2*|x}|=\sqrt[3]{-25*5}=\sqrt[3]{-125}=-5=x<0\).
Now, back to the original question:
If x<0, then \(\sqrt{-x*|x|}\) equals: A. \(-x\) B. \(-1\) C. \(1\) D. \(x\) E. \(\sqrt{x}\)
As square root function can not give negative result, then options -1 (B) and x (D) can not be the answers as they are negative. Also \(\sqrt{x}\) (E) can not be the answer as even root from negative number is undefined for GMAT. 1 (C) also can not be the answer as for different values of x the answer will be different, so it can not be some specific value. So we are left with A.
Now, if it were x>0 instead of x<0 then the question would be flawed as in this case the expression under the even root would be negative (-x*|x|=negative*positive=negative).
Re: Square root and Modulus [#permalink]
09 Dec 2010, 04:36
1
This post was BOOKMARKED
Friends,
In GMAT, to avoid the confusion with the negative #s, i use my own and simple technique that is as below.
PLEASE NOTE THAT: DEALING WITH THE INEQUALITIES/SQUARE-ROOTS/MODULUS THAT CONTAIN +VE #s IS EASIER THAN DEALING WITH THOSE CONTAIN -VE#S.
Given: x is a -ve I take: y as a +ve #. now i write -x = y
FROM NOW ON I AM GONNA DEAL WITH THE +VE # (i.e.y) ONLY. I CAN NOT BE TRAPPED BY THE TESTMAKER.
qtn: \(\sqrt{-x*|x|}\) = ? ==> \(\sqrt{y*|-y|}\) ==> \(\sqrt{y*y}\) as |-#|=# ==> \(\sqrt{y^2}\) ==> \(y\) (but not -y as Y is a positive #) ==> \(-x\) Answer "A"
Re: Square root and Modulus [#permalink]
30 Dec 2010, 14:20
Expert's post
marijose wrote:
I still do not understend why the x is negative. Is it because of the condition at the begining or because of the absolute value that is in the question?
here is whay I see. SQ ( -(-x)) *|-x|
you get... sq(x*x) therefore the answer is x, do you add the negative sign because the problem starts out saying that x is negative or is there a rule I am not seeing?
I was under the impression that the square root of any number cannot by negative or is this incorrect?
regards
x is negative because it's given in the stem that it's negative (if x<0...). Next, knowing that x is negative doesn't mean that you should write -x instead of x.
As for the square root: yes, even roots can not give negative result, so when you say that \(\sqrt{-x*|x|}\) equals to x, you are basically saying that square root of an expression equals to the negative number (as x is negative) which can not be true. Now, the answer is -x or -(negative)=positive, so square root of an expression equals to positive number which can be true. Also \(\sqrt{x^2}\) equals to |x| not to x. It's all explained above, so please read the solutions again and ask if anything needs further clarification.
Re: Square root and Modulus [#permalink]
30 Dec 2010, 14:47
Expert's post
marijose wrote:
Ive read the explanations .. sorry to keep bugging but
lets say you plug in a a -5. by doing all the calculations you get that the SQ(-(-5)*|-5|) = sq(5*5) = 5 so because and only because the question states that x is negative we have to change the value to -5. I this the correct reasoning?
I'm not sure understood your question.
What do you mean "change the value to -5"? If \(x=-5<0\) then \(\sqrt{-x*|x|}=\sqrt{-(-5)*|-5|}=\sqrt{25}=5\) so the answer is 5 or \(-x\), as \(-x=-(-5)=5\).
Re: Square root and Modulus [#permalink]
31 Dec 2010, 08:06
Expert's post
shrive555 wrote:
marijose wrote:
Ive read the explanations .. sorry to keep bugging but
lets say you plug in a a -5. by doing all the calculations you get that the SQ(-(-5)*|-5|) = sq(5*5) = 5 so because and only because the question states that x is negative we have to change the value to -5. I this the correct reasoning?
you don't have to plug in -5. Just Plug in 5, keep in mind that x<0 so it will make 5 as -5 automatically. so when you get the final answer after taking square root of 25 = 5. due to given condition of x<0 . you have to pick -5 or (negative value)from the answer choices. The only trick is " x<0 "
i was stuck in the same point
What do you mean by "it will make 5 as -5 automatically"???
And again: given that \(x<0\). If you want to solve this question by number plugging method you should plug negative values of \(x\) --> if \(x=-5<0\) then \(\sqrt{-x*|x|}=\sqrt{-(-5)*|-5|}=\sqrt{25}=5\) so the answer is 5 or \(-x\), as \(-x=-(-5)=5\). _________________
Re: Square root and Modulus [#permalink]
31 Dec 2010, 10:33
Bunuel wrote:
shrive555 wrote:
marijose wrote:
Ive read the explanations .. sorry to keep bugging but
lets say you plug in a a -5. by doing all the calculations you get that the SQ(-(-5)*|-5|) = sq(5*5) = 5 so because and only because the question states that x is negative we have to change the value to -5. I this the correct reasoning?
you don't have to plug in -5. Just Plug in 5, keep in mind that x<0 so it will make 5 as -5 automatically. so when you get the final answer after taking square root of 25 = 5. due to given condition of x<0 . you have to pick -5 or (negative value)from the answer choices. The only trick is " x<0 "
i was stuck in the same point
What do you mean by "it will make 5 as -5 automatically"???
And again: given that \(x<0\). If you want to solve this question by number plugging method you should plug negative values of \(x\) --> if \(x=-5<0\) then \(\sqrt{-x*|x|}=\sqrt{-(-5)*|-5|}=\sqrt{25}=5\) so the answer is 5 or \(-x\), as \(-x=-(-5)=5\).
Since x=5 but x<0 so x=-5 probably marijose is confusing the negative sign - with negative x .... \(\sqrt{-(x)*|x|}\) _________________
Re: Square root and Modulus [#permalink]
31 Dec 2010, 11:31
Bunuel wrote:
marijose wrote:
I still do not understend why the x is negative. Is it because of the condition at the begining or because of the absolute value that is in the question?
here is whay I see. SQ ( -(-x)) *|-x|
you get... sq(x*x) therefore the answer is x, do you add the negative sign because the problem starts out saying that x is negative or is there a rule I am not seeing?
I was under the impression that the square root of any number cannot by negative or is this incorrect?
regards
x is negative because it's given in the stem that it's negative (if x<0...). Next, knowing that x is negative doesn't mean that you should write -x instead of x.
This is what i meant but messed up in mathematics lingo _________________
Re: Is the square root always positive? [#permalink]
01 Feb 2011, 05:23
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Merging similar topics.
karenhipol wrote:
I encountered this question in gmat prep:
If x is < 0, what is the Square root of -x/x/
a. 0 b. 1 c. x d. -x e. -1
The official answer is -x but I answered x because I thought the answer should be positive.
Should I consider however, that since x <0 then I should multiply it by -1 to get a positive number so the answer is - x ( to get -(-x) = +x.
Please let me know If I reasoned this out correctly.
Thanks
Two things:
1. Square root function can not give negative result: \(\sqrt{some \ expression}\geq{0}\) so square root of a non-negative number is not always positive, it's never negative (always non-negative).
2. As \(x<0\) then \(-x=-negative=positive\) so the answer is still positive.
For the complete solution refer to the above posts. _________________
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