Power and roots confusion!!

[Jonna89]

Hi guys! This is my first post in my journey to beat the GMAT! Could you guys please help me understand the below question?

If N is positive, which of the following is equal to 1/(√N+1) - √N?

A. 1
B. (√2N)+1
C. (√N+1)/(√N)
D. (√N)+1-(√N)
E. (√N)+1+(√N)

I can solve most of the equation until I get to {(√N+1)+(√N)}/(N+1)-(√N). The answer is E,but I cannot understand/rationalize how the equation goes from {(√N+1)+(√N)}/(N+1)-(√N) to a result of (√N)+1+(√N). Thank you in advanced!!!

Best,
Jonna!

[Brent@GMATPrepNow]

Hi guys! This is my first post in my journey to beat the GMAT! Could you guys please help me understand the below question?

If N is positive, which of the following is equal to 1/(√N+1) - √N?

A. 1
B. (√2N)+1
C. (√N+1)/(√N)
D. (√N)+1-(√N)
E. (√N)+1+(√N)

I can solve most of the equation until I get to {(√N+1)+(√N)}/(N+1)-(√N). The answer is E,but I cannot understand/rationalize how the equation goes from {(√N+1)+(√N)}/(N+1)-(√N) to a result of (√N)+1+(√N). Thank you in advanced!!!

Best,
Jonna!

I have a question about the notation in your question.
You write 1/(√N+1).
Are we taking the square root of N or N+1?

If we're taking the square root of N+1, it should read: 1/√(N+1).

Cheers,
Brent

[Jonna89]

Hi Brent. You are correct, it is √(N+1).The question should read: 1/√(N+1) - (√N). The correct answer is E √(N+1) + √N. Please advise. Thanks!

[[email protected]]

Hi Jonna89,

Has the entire prompt (and the answer choices) been properly transcribed? If you can upload a picture of the original question (and answers), then that would probably make all of this easier.

GMAT assassins aren't born, they're made,
Rich

[pdvbhat]

You just need to rationalize, by multiplying the numerator and denominator by (√(N+1) + √N) to get option E, as the denominator will be equal to 1 after rationalization i.e.
(√(N+1)-√N)(√(N+1)+√N)= N+1-N = 1.

[GMATGuruNY]

Let n = 2.
Then:
1/(√n+1 - √n) = 1/(√3-√2) ≈ 1/(1.7 - 1.4) = 1/.3 = 10/3. This is our target.

Now plug n=2 into the answers to see which comes closest to our target of 10/3.

Only E works:
√(n+1) + √n = √3+√2 ≈ 1.7 + 1.4 = 3.1.

The correct answer is E.

[Matt@VeritasPrep]

Here's the basic idea, in more legible form: check it out! I tried to make it as thorough as I could.



There's a typo on the screenshot -- (√n+1 - √n)² should give n+1 - 2√n+1√n + n -- but other than that, all should be well.

[Jonna89]

Wow! That actually made so much sense! You guys are awesome!! Thank you! I wish I would have join the site earlier!!! Cheers!

[Matt@VeritasPrep]

Wow! That actually made so much sense! You guys are awesome!! Thank you! I wish I would have join the site earlier!!! Cheers!

Better late than never! Keep up the good work.

[Brent@GMATPrepNow]

In mathematics, it's generally considered poor form to write fractions with roots in the denominator (although having roots in the numerator is acceptable). So, to eliminate the roots in the denominator, we must multiple both numerator and denominator by something that accomplishes this.

Here that something is (√(N+1) + √N).

Aside: (√(N+1) + √N) is known as the "conjugate" of (√(N+1) - √N)

Here's another question that tests this concept: https://www.beatthegmat.com/help-roots-q-t271220.html

Cheers,
Brent