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Rationalizing a Denominator with a Binomial - Concept

Teacher/Instructor Carl Horowitz
Carl Horowitz

University of Michigan
Runs his own tutoring company

Carl taught upper-level math in several schools and currently runs his own tutoring company. He bets that no one can beat his love for intensive outdoor activities!

When rationalizing a denominator with two terms, called a binomial, first identify the conjugate of the binomial. The conjugate is the same binomial except the second term has an opposite sign. Next, multiply the numerator and denominator by the conjugate. The denominator becomes a difference of squares, which will eliminate the square roots in the denominator.

Rationalizing the denominator with a binomial with 2 terms in the denominator. So for this example what we're looking at 3 over 1 plus root 3, and we want to rationalize the denominator. We want to get rid of that square root in the denominator.
What we're used to doing is just multiplying by the square root that we have, okay. So what I mean by that is if we multiply this by root 3 over root 3, what actually ends up happening is this root 3 gets distributed into both things, so we end up with root 3 plus root 3 times root 3 which is just going to be 3. We haven't actually gotten rid of our square root at all. All we've done is move the square root from one term to the other and made our terms bigger, okay? So this isn't going to work.
What we actually have to do is multiply by something else, and that something else is called the conjugate which is basically the same numbers but the opposite sign in between. So that in this case is going to be one minus root 3 over one minus root 3, it has to be multiplied by 1. And what happens is now when we multiply this through we have to foil it out and so we end up with 1. We have positive root 3 and negative root 3 which cancel, so those disappear and then we have root 3 times root 3 which is 3. This should be our negative. So we end up with 1-3 in the denominator or -2 which is what we wanted. We wanted to get rid of that negative sign from the denominator. Okay?
We still have to distribute this 3 in the numerator. So we end up with 3-3 root 3 all over -2. Okay?
So when we're multiplying, sorry. In order to get rid of the square root from the denominator when we have two terms, we have to multiply by the conjugate which is the same thing but just with a different sign in between.

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