(1+sqrt(2)+sqrt(3))/(1-sqrt(2)+sqrt(3))
=(1+sqrt(2)+sqrt(3))^2/[(1-sqrt(2)+sqrt(3))(1+sqrt(2)+sqrt(3))]
=(1+2+3+2sqrt(2)+2sqrt(2)+2sqrt(6))/((sqrt(3)+1)^2-2)
=(3+sqrt(2)+sqrt(2)+sqrt(6))/(sqrt(3)+1)
=(3+sqrt(2)+sqrt(2)+sqrt(6))(sqrt(3)-1)/[(sqrt(3)+1)(sqrt(3)-1)]
=(2sqrt(2)+2sqrt(3))/2
=sqrt(2)+sqrt(3)
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