\(\left(\dfrac{2}{\sqrt{3}-1}+\dfrac{3}{\sqrt{3}-2}+\dfrac{15}{3-\sqrt{3}}\right).\dfrac{1}{\sqrt{3}+5}\\ =\left(\dfrac{2.\sqrt{3}\left(\sqrt{3}-2\right)}{\sqrt{3}\left(\sqrt{3}-1\right)\left(\sqrt{3}-2\right)}+\dfrac{3\sqrt{3}\left(\sqrt{3}-1\right)}{\left(\sqrt{3}-2\right)\sqrt{3}\left(\sqrt{3}-1\right)}+\dfrac{15\left(\sqrt{3}-2\right)}{\left(\sqrt{3}-2\right)\sqrt{3}\left(\sqrt{3}-1\right)}\right).\dfrac{1}{\sqrt{3}+5}\\ =\dfrac{-15+8\sqrt{3}}{\left(\sqrt{3}-2\right)\sqrt{3}\left(\sqrt{3}-1\right)}.\dfrac{1}{\sqrt{3}+5}\\ =\dfrac{\sqrt{3}\left(8-5\sqrt{3}\right)}{\left(\sqrt{3}-2\right)\sqrt{3}\left(\sqrt{3}-1\right)}.\dfrac{1}{\sqrt{3}+5}\\ =\dfrac{8-5\sqrt{3}}{\left(5-3\sqrt{3}\right)\left(\sqrt{3}+5\right)}=\dfrac{8-5\sqrt{3}}{16-10\sqrt{3}}=\dfrac{1}{2}\)
\(\left(\dfrac{2}{\sqrt{3}-1}+\dfrac{3}{\sqrt{3}-2}+\dfrac{15}{3-\sqrt{3}}\right)\cdot\dfrac{1}{\sqrt{3}+5}\)
\(=\left(\dfrac{2\left(\sqrt{3}+1\right)}{2}+\dfrac{3\left(\sqrt{3}+2\right)}{-1}+\dfrac{15\left(3+\sqrt{3}\right)}{6}\right)\cdot\dfrac{1}{\sqrt{3}+5}\)
\(=\left(\sqrt{3}+1-3\left(\sqrt{3}+2\right)+\dfrac{5\left(3\sqrt{3}\right)}{2}\right)\cdot\dfrac{1}{\sqrt{3}+5}\)
\(=\left(\sqrt{3}+1-3\sqrt{3}-6+\dfrac{15+5\sqrt{3}}{2}\right)\cdot\dfrac{1}{\sqrt{3}+5}\)
\(=\left(-2\sqrt{3}-5+\dfrac{15+5\sqrt{3}}{2}\right)\cdot\dfrac{1}{\sqrt{3}+5}\)
\(=\left(\dfrac{-4\sqrt{3}+15+5\sqrt{3}}{2}-5\right)\cdot\dfrac{1}{\sqrt{3}+5}\)
\(=\left(\dfrac{\sqrt{3}+15}{2}-5\right)\cdot\dfrac{1}{\sqrt{3}+5}\)
\(=\dfrac{\sqrt{3}+15-10}{2}\cdot\dfrac{1}{\sqrt{3}+5}\)
\(=\dfrac{\sqrt{3}+5}{2}\cdot\dfrac{1}{\sqrt{3}+5}\)
\(=\dfrac{1}{2}\)
