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[latex] frac{1}{sqrt3+sqrt2+1} = frac{sqrt3+sqrt2-1}{(sqrt3+sqrt2+1)(sqrt3+sqrt2-1)} =frac{sqrt3+sqrt2-1}{(sqrt3+sqrt2)^2-1} = frac{sqrt3+sqrt2-1}{5+2sqrt6-1}=\\=frac{sqrt3+sqrt2-1}{2sqrt2(sqrt3+sqrt2)} = frac{(sqrt3+sqrt2-1)(sqrt3-sqrt2)}{2sqrt2(sqrt3+sqrt2)(sqrt3-sqrt2)}= frac{(sqrt3+sqrt2)(sqrt3-sqrt2)-(sqrt3-sqrt2)}{2sqrt2(3-2)} =[/latex][latex]= frac{3-2-sqrt3+sqrt2}{2sqrt2}=frac{1+sqrt2-sqrt3}{2sqrt2} =frac{(1+sqrt2-sqrt3)sqrt2}{2sqrt2cdot sqrt2}=frac{sqrt2+2-sqrt6}{4 } [/latex]