SOLUTION: How would I do this problem? Express 5 over 3-√2 with a rational denominator, in simplest radical form.

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Question 275432: How would I do this problem? Express 5 over 3-√2 with a rational denominator, in simplest radical form.
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
5%2F%283-sqrt%282%29%29
Your denominator has two terms. To rationalize a denominator of two terms we take advantage of the pattern: %28a%2Bb%29%28a-b%29+=+a%5E2+-+b%5E2. Every two-term expression (aka binomial) will match one of the two factors on the left side of the pattern. In your case, because you have a "-" between the two terms of your denominator, your denominator matches the (a-b) factor in the pattern. If we multiply your denominator by 3%2Bsqrt%282%29 we will get, as we can see from the pattern, two perfect squares (with a "-" between them). This is exactly what you need to rationalize the denominator.

Of course we cannot multiply just the denominator by 3%2Bsqrt%282%29. We must also multiply the numerator by the same thing. Otherwise we are changing the value of the expression:
%285%2F%283-sqrt%282%29%29%29%28%283%2Bsqrt%282%29%29%2F%283%2Bsqrt%282%29%29%29
On top we use the Distributive Property. On the bottom, the pattern tells us what we will get:
%2815%2B5sqrt%282%29%29%2F%283%5E2-%28sqrt%282%29%29%5E2%29
which simplifies as follows:
%2815%2B5sqrt%282%29%29%2F%289+-+2%29
%2815%2B5sqrt%282%29%29%2F7
and the denominator is now rational.