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}}$. Rationalize the denominator: frac{1}{1 + \sqrt{2} - \sqrt{3}}$. 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 $a + b + c$?

May 23, 2019

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$$\frac{\sqrt{2} + a + \sqrt{b}}{c}$$

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

May 24, 2019