Question: Root in the Denominator

Comment on Root in the Denominator

I solved this in a slightly different way,and just wanted to check if you could tell me which one would work faster in multiple scenarios.
So instead of trying to get the same denominator as 1 by using two conjugates, I multiplied both the denominators with each other to get a common multiple in the denominator, and then in the numerators, I multiplied the first fraction's numerator by the second denominator, and similarly for the other one.

Do let me know.
Thanks
gmat-admin's picture

That works too.

Your method works because the each of the two denominators just happens to be the conjugate of the other.

Hey!
Thanks for the response.
Just a follow up question. Will my method not work, in case they are not conjugates?
gmat-admin's picture

Yes, yours is still a valid approach, but you may have to perform a little extra work (not much though).

hi brent,

so if the right part of the question is the conjugate of the left part then we don't have to solve for it as we can assume that it would be the opposite of what we get on the left part of the question.

in other words, we got -1 + root 3/1 for the left part so is it safe to assume that the right part will be -1 - root 3/1
gmat-admin's picture

Sorry, but I'm not 100% sure what you're asking when you write "we can assume that it would be the opposite of what we get on the left part of the question."

Can you please elaborate?

Aside: If A is the conjugate of B, then we also know that B is the conjugate of A.
For example, 5 - √3 is the conjugate of 5 + √3, and we can also say that 5 + √3 is the conjugate of 5 - √3

what I mean is, for the question:

1 + √3 1 - √3
------ - ------
2 + √3 2 - √3

In the first step of your solution:

1 + √3
------
2 + √3


you multiplied the denominator and the numerator by 2 - √3 to remove the root. And by solving you got the answer:-1+√3
-----
1
so I asked whether it was necessary to perform the same operations to the other fraction as the answer will also be the conjugate since the two fractions are conjugate of each other.
gmat-admin's picture

Thanks for clarifying.
You're absolutely right; since each part of the second fraction is the conjugate of each part of the first fraction, the resulting numerator (after we multiply top and bottom by their respective conjugates) of the second fraction will be the conjugate of the numerator of the first fraction.

Cheers,
Brent

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