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If we write sqrt(2) + sqrt(3) + 1/(2*sqrt(2) + 3*sqrt(3)) in the form (a*sqrt(2) + b*sqrt(3))/c such that a, b, and c are positive integers and c is as small as possible, then what is a + b + c?

 Feb 25, 2022
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Rationalize the denominator of the term:  1 / [ 2·sqrt(2) + 3·sqrt(3) ] 

by multiplying both the numerator and denominator by the conjugate of the 

denominator which is:  2·sqrt(2) - 3·sqrt(3).

 

1 / [ 2·sqrt(2) + 3·sqrt(3) ]    ·    [ 2·sqrt(2) - 3·sqrt(3) ] / [ 2·sqrt(2) - 3·sqrt(3) ] 

      =     [ 2·sqrt(2) - 3·sqrt(3) ] / -19     =    [ 3·sqrt(3) -  2·sqrt(2) ] / 19

 

Rewrite this as two separate fractions:  3·sqrt(3) / 19  -  2·sqrt(2) / 19

 

Now, write the other two terms with a donominator of 19:

   sqrt(2)  =  19·sqrt(2) / 19          and         sqrt(3)  =  19·sqrt(3) / 19

 

Then, simplify this:  19·sqrt(2) / 19  +  19·sqrt(3) / 19  +  3·sqrt(3) / 19  -  2·sqrt(2) / 19

     =    17·sqrt(2) / 19  +  22·sqrt(3) / 19

     =     [ 17·sqrt(2) / 19  +  22·sqrt(3) ] / 19

 

From this, you can find the answer.

 Feb 25, 2022

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