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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?

 Jul 5, 2021
 #1
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First:  write  1 / [ 2sqrt(2) + 3sqrt(3) ] with a rational denominator by multiplying both

          the numerator and the denominator by the conjugate of the denominator: 

                   2sqrt(2) - 3sqrt(3):

 

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

 

=   [ 2sqrt(2) - 3sqrt(3) ] / [ 8 - 27 ]

=   [ 2sqrt(2) - 3sqrt(3) ] / - 19

=   [ 3sqrt(3) - 2sqrt(2) / 19

 

Second:  wrtie  sqrt(2) + sqrt(3)  with a denominator of 19:

=   19sqrt(2) / 19  +  19sqrt(3) / 19

 

Add together these two results:  [ 3sqrt(3) - 2sqrt(2) / 19  +  19sqrt(2) / 19  +  19sqrt(3) / 19

 

=  [ 22sqrt(3) + 17sqrt(2) ] / 19

 

a  =  17     b = 22     c  =  19

 Jul 5, 2021

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