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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 \(\sqrt{-x*|x|}=\sqrt{positive*positive}=\sqrt{positive}\).

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 cannot 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).

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"

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.

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 \(-x=-(-5)=5\).

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 \(-x=-(-5)=5\).
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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 .... \(\sqrt{-(x)*|x|}\)
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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
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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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