Rationalizing Denominators

BREAKING: Target Test Prep releases Brand New 2026 On Demand GMAT prep course

Redeem

This topic has expert replies

Post new topic Post Reply

Previous Topic Next Topic

NYC493: Junior | Next Rank: 30 Posts; Posts: 25; Joined: Tue Apr 05, 2011 5:45 am; Location: New York City; Followed by:1 members

Rationalizing Denominators

by NYC493 » Fri Mar 23, 2012 8:47 am

Hi everyone,

I'm not so good at simplifying complicated expressions. I see the the OG 12th Edition makes a few references to the method of "rationalizing the denominator". An example of this is OG Problem Solving #117, answer explanation on pg. 223:

#117) If n is positive, which of the following is equal to 1/sqrt(n+1) - sqrt(n)?

OG goes on to explain that the denominator can be rationalized by multiplying the expression by sqrt(n+1) + sqrt(n)/ sqrt(n+1) + sqrt(n).

Can anyone explain in plain english what it means to "rationalize the denominator", and why it's useful? In the case of #117, why is it achieved by multiplying sqrt(n+1) - sqrt(n) by sqrt(n+1) + sqrt(n)? Seems like there's a trick that I don't know about.

Thanks!!!

Quote

Source: Beat The GMAT — Problem Solving | Fri Mar 23, 2012 8:47 am

GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pm; Location: New York, NY; Thanked: 13060 times; Followed by:1906 members; GMAT Score:790

by GMATGuruNY » Fri Mar 23, 2012 9:23 am

NYC493 wrote:Hi everyone,

I'm not so good at simplifying complicated expressions. I see the the OG 12th Edition makes a few references to the method of "rationalizing the denominator". An example of this is OG Problem Solving #117, answer explanation on pg. 223:

#117) If n is positive, which of the following is equal to 1/sqrt(n+1) - sqrt(n)?

OG goes on to explain that the denominator can be rationalized by multiplying the expression by sqrt(n+1) + sqrt(n)/ sqrt(n+1) + sqrt(n).

Can anyone explain in plain english what it means to "rationalize the denominator", and why it's useful? In the case of #117, why is it achieved by multiplying sqrt(n+1) - sqrt(n) by sqrt(n+1) + sqrt(n)? Seems like there's a trick that I don't know about.

Thanks!!!

To RATIONALIZE a fraction is to remove a âˆš from the denominator.
To remove a âˆš from the denominator, we take advantage of the DIFFERENCE OF TWO SQUARES:
(x+y)(x-y) = xÂ² - yÂ².

(x+y) and (x-y) are CONJUGATES of each other.
If either (x+y) or (x-y) is in the denominator and has a âˆš, the âˆš can be removed by multiplying by the conjugate.
In order not to change the value of the fraction, we multiply BOTH THE TOP AND THE BOTTOM by the conjugate.
(Multiplying the top and the bottom by the same value is the same as multiplying by 1, leaving the value of the fraction unchanged.)

In the problem at hand, we must rationalize 1 / (âˆš(n+1) - âˆšn).
The conjugate of (âˆš(n+1) - âˆšn) is (âˆš(n+1) + âˆšn).
Multiplying both the top and the bottom by the conjugate, we get:

1 / (âˆš(n+1) - âˆšn)

= 1 * (âˆš(n+1) + âˆšn) / (âˆš(n+1) - âˆšn)(âˆš(n+1) + âˆšn)

= (âˆš(n+1) + âˆšn) / (âˆšn+1)Â² - (âˆšn)Â²

= (âˆš(n+1) + âˆšn) / (n+1) - n

= (âˆš(n+1) + âˆšn) / 1

= âˆš(n+1) + âˆšn.

I offered an alternate solution to this problem here:

https://www.beatthegmat.com/tough-ps-t94365.html

Private tutor exclusively for the GMAT and GRE, with over 20 years of experience.
Followed here and elsewhere by over 1900 test-takers.
I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.

As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.

For more information, please email me (Mitch Hunt) at [email protected].
Student Review #1
Student Review #2
Student Review #3