}}$. Rationalize the denominator: frac{1}{1 + \sqrt{2} - \sqrt{3}}$. The simplified result can be written in the form $\frac{\sqrt{2} + a + \sqrt{b}}{c}$, where $a$, $b$, and $c$, are positive integers. What is $a + b + c$?
\($\frac{\sqrt{2} + a + \sqrt{b}}{c}$ \)
1 [ 1 - ( √2 - √3) ] 1 - √2 + √3 1 - √2 + √3 1 - √2 + √3
__________________________ = ______________ = _____________ = ________ =
[1 + (√2 - √3)] [ 1 - ( √2 - √3) ] 1 - (√2 - √3)^2 1 - [ 2 - 2√6 +3] 2√6 - 4
1 - √2 + √3
_________ =
2(√6 - 2)
[1 - √2 + √3 ] ( √6 + 2) [ √6 - √12 + √18 + 2 - 2√2 + 2√3 ]
__________________ = ____________________________ =
2(√6 - 2) (√6 + 2) 2 ( 6 - 4)
[ √6 - 2√3 + 3√2 + 2 - 2√2 + 2√3 ]
__________________________ =
2 (6 - 4)
√2 + 2 + √6
__________
4
So
a + b + c = 2 + 6 + 4 = 12