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

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

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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

We rationalize the denominator by multiplying the expression by [√(n - 1) + √(n + 1)]/[√(n - 1) + √(n + 1)], and we have:

-2[√(n - 1) + √(n + 1)]/[√(n - 1)^2 - √(n + 1)^2]

-2[√(n - 1) + √(n + 1)]/[n - 1 - (n + 1)]

-2[√(n - 1) + √(n + 1)]/-2 = √(n - 1) + √(n + 1)

Answer: D

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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