SOLUTION: Rationalize the denominator of 7√3-5√2 divided by √48+√18

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Question 617255: Rationalize the denominator of 7√3-5√2 divided by √48+√18
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
%287sqrt%283%29-5sqrt%282%29%29%2F%28sqrt%2848%29%2Bsqrt%2818%29%29
Before going about rationalizing the denominator, I'm going to simplify the denominator. It will make the numbers smaller and easier to work with.
%287sqrt%283%29-5sqrt%282%29%29%2F%28sqrt%2816%2A3%29%2Bsqrt%289%2A2%29%29

%287sqrt%283%29-5sqrt%282%29%29%2F%284sqrt%283%29%2B3sqrt%282%29%29

Now we'll rationalize. Rationalizing two-term denominators with square roots takes advantage of the %28a%2Bb%29%28a-b%29+=+a%5E2-b%5E2 pattern. The right side, as you can see is a difference of perfect squares. The left side is a product of two-term expressions. The pattern shows us how to take a two-term expression, an (a+b) or (a-b), and turn it into an expression of perfect squares.

Your denominator is a two-term sum. IOW, an (a+b). To turn it into an expression of perfect squares we need to multiply it by its corresponding (a-b):

Multiplying the denominators is easy; just use the pattern. To multiply the numerators we will need to use FOIL:

which simplifies as follows:
%2828%2A3%2B%28-21sqrt%286%29%29%2B%28-20sqrt%286%29%29%2B15%2A2%29%2F%2816%2A3-9%2A2%29
%2884%2B%28-41sqrt%286%29%29%2B30%29%2F%2848-18%29
%28114%2B%28-41sqrt%286%29%29%29%2F30