[tex]\boxed{\begin{array}{l}\sf 1+\dfrac{2}{1+\sqrt{3}} +\dfrac{1}{2+\sqrt{3}}\\\sf \dfrac{(1+\sqrt{3})(2+\sqrt{3})+2+\sqrt{3}+1+\sqrt{3}}{(1+\sqrt{3})(2+\sqrt{3})}\\\sf =\dfrac{2+\sqrt{3}+2\sqrt{3}+3+2+\sqrt{3}+1+\sqrt{3}}{(1+\sqrt{3})(3+\sqrt{3})}\\\sf\dfrac{8+5\sqrt{3}}{5+3\sqrt{3}}\\\sf\dfrac{(8+5\sqrt{3})}{(5+3\sqrt{3})}\cdot\dfrac{(5-3\sqrt{3})}{(5-3\sqrt{3})}\\\sf=\dfrac{40-24\sqrt{3}+25\sqrt{3}-15(\sqrt{3})^2}{5^2-(\sqrt{3})^2}\\\sf=\dfrac{40-24\sqrt{3}+25\sqrt{3}-45}{25-3}=\dfrac{-5+\sqrt{3}}{22}\end{array}}\blue{\checkmark}[/tex]
