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?
+129850 sqrt (2) + sqrt (3) + 1 / [ (2sqrt (2) + 3sqrt (3) ]
[(sqrt (2) + sqrt (3) ) ( 2sqrt ( 2) + 3 sqrt (3) ) + 1 ] / ( 2sqrt (2) + 3sqrt (3) )
[ 4 + 5sqrt (6) + 9 + 1 ] / ( sqrt (8) + sqrt (27))
[ 14 + 5sqrt (6) ] / ( sqrt 8 + sqrt (27) mult num/den by sqrt (8) - sqrt (27)
[14 + 5sqrt (6) [ sqrt (8) - sqrt (27)] / ( 8 - 27)
[14sqrt (8) + 5 sqrt (48) - 14sqrt(27) - 5sqrt (162) ] / (-19)
[ 14 * 2sqrt (2) + 5*4 sqrt (3) - 14*3 sqrt (3) - 5*9 sqrt ( 2) ] / (-19)
[ -17 sqrt 2 - 22 sqrt 3 ] / (-19)
[ 17 sqrt 2 + 22 sqrt (3) ] / 19
a = 17 b = 22 c =19
a + b + c = 58
+26388 If we write \(\sqrt{2} + \sqrt{3} + \dfrac{1}{ (2*\sqrt{2} + 3*\sqrt{3}) }\)
in the form \(\dfrac{(a*\sqrt{2} + b*\sqrt{3})}{c}\) such that a, b, and c
are positive integers and c is as small as possible,
then what is a + b + c?
\(\begin{array}{|rcll|} \hline && \sqrt{2} + \sqrt{3} + \dfrac{1}{ (2*\sqrt{2} + 3*\sqrt{3}) } \\\\ &=& \sqrt{2} + \sqrt{3} + \dfrac{1}{ (2*\sqrt{2} + 3*\sqrt{3}) } * \dfrac{(2*\sqrt{2} - 3*\sqrt{3})}{ (2*\sqrt{2} - 3*\sqrt{3}) } \\\\ &=& \sqrt{2} + \sqrt{3} + \dfrac{(2*\sqrt{2} - 3*\sqrt{3})}{ (2*\sqrt{2} + 3*\sqrt{3})(2*\sqrt{2} - 3*\sqrt{3}) } \\\\ &=& \sqrt{2} + \sqrt{3} + \dfrac{(2*\sqrt{2} - 3*\sqrt{3})}{ (4*2-9*3) } \\\\ &=& \sqrt{2} + \sqrt{3} + \dfrac{(2*\sqrt{2} - 3*\sqrt{3})}{ -19 } \\\\ &=& \sqrt{2} + \sqrt{3} + \dfrac{-(2*\sqrt{2} - 3*\sqrt{3})}{ 19 } \\\\ &=& \dfrac{19*(\sqrt{2} + \sqrt{3}) -(2*\sqrt{2} - 3*\sqrt{3})}{ 19 } \\\\ &=& \dfrac{19*\sqrt{2} + 19*\sqrt{3} -2*\sqrt{2} + 3*\sqrt{3}}{ 19 } \\\\ &=& \dfrac{{\color{red}17}*\sqrt{2} + {\color{red}22}*\sqrt{3}}{ {\color{red}19} } \\ \hline \end{array}\)
\(\begin{array}{|rcll|} \hline a+b+c &=& 17+22+19 \\ \mathbf{a+b+c} &=& \mathbf{58} \\ \hline \end{array}\)
