SOLUTION: I need steps for simplifying 10/(sqrt7+sqrt2) Thank you!!

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Question 290054: I need steps for simplifying 10/(sqrt7+sqrt2)
Thank you!!
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
10%2F%28sqrt%287%29%2Bsqrt%282%29%29
The denominator has two terms. To rationalize this denominator we need to find a way to take a two-term expression and change it into an expression of nothing but perfect squares. Since multiplying the numerator and denominator is the primary way to change a denominator, we are looking for an expression we can multiply your denominator by to get an expression of perfect squares.

The key to this is the pattern:
%28a%2Bb%29%28a-b%29+=+a%5E2+-+b%5E2
On the left we have two two-term expressions being multiplied. And on the right we have an expression of perfect squares! This can be used to rationalize any two-term denominator with 1 or 2 square roots.

Your two-term denominator has a "+" between the two terms. So your denominator is the (a+b). We need to multiply by (a-b):

%2810sqrt%287%29-10sqrt%282%29%29%2F%28%28sqrt%287%29%29%5E2-%28sqrt%282%29%29%5E2%29%29
%2810sqrt%287%29-10sqrt%282%29%29%2F%287-2%29
%2810sqrt%287%29-10sqrt%282%29%29%2F5
%285%282sqrt%287%29-2sqrt%282%29%29%29%2F5
%28cross%285%29%282sqrt%287%29-2sqrt%282%29%29%29%2Fcross%285%29
2sqrt%287%29-2sqrt%282%29

Not only did we rationalize the denominator, we eliminated the fraction entirely!