$\frac{6}{3+2\sqrt{3}}+\frac{6}{3-2\sqrt{3}}=\frac{6(3-2\sqrt{3})+6(3+2\sqrt{3})}{(3+2\sqrt{3})(3-2\sqrt{3})}=\frac{18-12\sqrt{3}+18+12\sqrt{3}}{3^2-(2\sqrt{3}{)}^2}=\frac{36}{9-4\ast 3}=\frac{36}{9-12}=\frac{36}{-3}=-12$ − рациональное число

$\frac{\sqrt{11}+\sqrt{6}}{\sqrt{11}-\sqrt{6}}+\frac{\sqrt{11}-\sqrt{6}}{\sqrt{11}+\sqrt{6}}=\frac{(\sqrt{11}+\sqrt{6}{)}^2+(\sqrt{11}-\sqrt{6}{)}^2}{(\sqrt{11}-\sqrt{6})(\sqrt{11}+\sqrt{6})}=\frac{(\sqrt{11}{)}^2+2\ast \sqrt{11}\ast \sqrt{6}+(\sqrt{6}{)}^2+(\sqrt{11}{)}^2-2\ast \sqrt{11}\ast \sqrt{6}+(\sqrt{6}{)}^2}{(\sqrt{11}{)}^2-(\sqrt{6}{)}^2}=\frac{11+2\sqrt{66}+6+11-2\sqrt{66}+6}{11-6}=\frac{34}{5}=6\frac{4}{5}$ − рациональное число