Which of the following is equal to -2/((n-1)^(1/2) - (n+1)^(1/2)) for

Question

Which of the following is equal to  \frac{-2}{\sqrt{n-1} - \sqrt{n+1}}  for all values of n > 1?

A. -1

B. 1

C.  2(\sqrt{n-1} + \sqrt{n+1})

D.  \sqrt{n-1} + \sqrt{n+1}

E.  \frac{\sqrt{n-1}}{\sqrt{n+1}}

u1983 · Accepted Answer

( \sqrt{n-1} - \sqrt{n+1} ) * (  \sqrt{n-1} + \sqrt{n+1} )  = n-1-n-1=-2 
Thus multiplying  \sqrt{n-1} + \sqrt{n+1}  at both N & D of the expression  \frac{-2}{\sqrt{n-1} - \sqrt{n+1}}  we get
 \frac{(-2)}{(-2)} * \sqrt{n-1} + \sqrt{n+1} 
= \sqrt{n-1} + \sqrt{n+1}  ....... option D .

aserghe1 · Answer

I just substituted n=2 into the expression and checked which answer choice gave the same result. Is there a better approach?

syedazeem3 · Answer

I wasn’t able to get the answer using plug in, is there a better way to do this?

Posted from my mobile device

aserghe1 · Answer

Since 1<n, let's plug n=2 into this expression:

\frac{-2}{\sqrt{n-1} - \sqrt{n+1}}---->\frac{-2}{\sqrt{2-1} - \sqrt{2+1}}

This simplifies to
 1+\sqrt{3}

Let's examine the answer choices.

A...  1+\sqrt{3}  does not equal -1, so this answer is eliminated

B...  1+\sqrt{3}  does not equal 1, so this answer is eliminated

C... Plug in n=2 (since that's what we used in the beginning) into  2(\sqrt{n-1} + \sqrt{n+1})---->2(1+\sqrt{3}) -----> this doesnt equal  1+\sqrt{3}  so answer is eliminated

D... Plug in n=2 into the answer choice.  \sqrt{2-1}+\sqrt{2+1}=1+\sqrt{3}  ---> this matches our original expression, so this is the answer

Answer: D

JeffTargetTestPrep · Answer

We rationalize the denominator by multiplying the expression by  / , and we have:

-2 /

-2 /-2 = √(n - 1) + √(n + 1)

Answer: D

CrackverbalGMAT · Answer

This is a question on surds/roots. When you have surds in the denominator of a fraction, you need to rationalize the denominator by multiplying AND dividing by the conjugate of the surd.

Rationalization is the process of converting an irrational number (like a surd) to a rational number by multiplying an dividing by its conjugate. When we do this, we obtain an expression, which is usually in the form of (a-b) * (a+b) which can then be simplified to  a^2 - b^2  followed by cancelling terms from the Numerator and Denominator.

JeffTargetTestPrep  &  u1983  have already highlighted this process very well in their replies.

The denominator is √(n-1) - √(n+1 which will always be an irrational number. To convert this to a rational number, we multiply it by its conjugate i.e. √(n-1) +√(n+1) . When we do this, we obtain (n-1)-(n+1) . Cancelling out n, we have a -2 in the denominator which gets cancelled out with the -2 in the numerator. 
We will be left with √(n-1) - √(n+1) in the numerator and hence  the correct answer option is D .

@aserghe, I personally advocate the plugging-in approach in a lot of cases and there’s absolutely nothing wrong with your approach. But, I want you to bear in mind that when you plug-in values, remember to plug in all sorts of values and not only integral values. Also remember to try two or three different values before you decide on your final answer; on the difficult questions, trying just one value may not be sufficient.

Hope that helps!

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Which of the following is equal to -2/((n-1)^(1/2) - (n+1)^(1/2)) for

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Which of the following is equal to -2/((n-1)^(1/2) - (n+1)^(1/2)) for

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