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!!!
Rationalizing Denominators
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To RATIONALIZE a fraction is to remove a √ from the denominator.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 remove a √ from the denominator, we take advantage of the DIFFERENCE OF TWO SQUARES:
(x+y)(x-y) = x² - y².
(x+y) and (x-y) are CONJUGATES of each other.
If either (x+y) or (x-y) is in the denominator and has a √, the √ 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 / (√(n+1) - √n).
The conjugate of (√(n+1) - √n) is (√(n+1) + √n).
Multiplying both the top and the bottom by the conjugate, we get:
1 / (√(n+1) - √n)
= 1 * (√(n+1) + √n) / (√(n+1) - √n)(√(n+1) + √n)
= (√(n+1) + √n) / (√n+1)² - (√n)²
= (√(n+1) + √n) / (n+1) - n
= (√(n+1) + √n) / 1
= √(n+1) + √n.
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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