This is a pretty ugly looking fraction because it has 2 square roots in the denominator. What I’m going to do to rationalize the denominator is multiply the top and bottom of this fraction by 1. It’s not just going to be the number 1 though, I’m going to multiply top and bottom by the conjugate of the denominator. Conjugate means it's going to have root 7 and root 10 still, only instead of a plus sign, it’s going to be a minus sign and here is why that’s clever.
Anytime you’re multiplying conjugates, you’re going to end up with an integer as your answer. Let me show you what I mean. Let’s look at the denominator and let’s FOIL. First, root 7 times root 7 is regular old 7. Outers, root 7 times root 10 is -70. Inners, I have + root 70 and then lasts I have -10, that was the Foiling on the bottom and then if you look those 2 terms cancel out, they’re additive inverses negative root 70 and positive root 70. So really on the bottom of my fraction I’m just going to have 7 take away 10 which is -3.
That was the whole point of this messy multiplication stuff by the way I’ll get back to the numerator in a second. All I wanted to show you here was the denominator. The reason why we multiplied by the conjugate is because instead of having these nasty square roots in the bottom, if we multiply by 1, we end up with -3 an integer in the denominator. That will always be the case if you multiply correctly by the conjugate.
Okay let’s go back to the top. Bottoms all done on top I needed to distribute this 4. So I’ll have 4 root 7 take away 4 root 10 and neither root 7 nor root 10 can be simplified any further, so I know that that’s my final answer.
So guys again if you see a product, excuse me, if you see a sum or difference of radicals in the bottom of a fraction, what you’re going to be doing is just multiplying by the conjugate of the denominator business. Make sure you change that sign. If it was a plus sign, you make it a minus sign here. After that it’s pretty straight forward, FOIL correctly, don’t lose any of your positive or negative signs and you guys will get the right answer.
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