Question

(\(\frac{2}{\sqrt{3}-1}+\frac{3}{\sqrt{3}-2}+\frac{15}{3-\sqrt{3}}\)).\(\frac{1}{\sqrt{3}+5}\)

Answer 1

\(=\left(\frac{2\left(\sqrt{3}+1\right)}{\left(\sqrt{3}-1\right)\left(\sqrt{3}+1\right)}+\frac{3\left(\sqrt{3}+2\right)}{\left(\sqrt{3}-2\right)\left(\sqrt{3}+2\right)}+\frac{15\left(3+\sqrt{3}\right)}{\left(3+\sqrt{3}\right)\left(3-\sqrt{3}\right)}\right)\frac{1}{5+\sqrt{3}}\)

\(=\left(\sqrt{3}+1-3\left(\sqrt{3}+2\right)+\frac{5\left(3+\sqrt{3}\right)}{2}\right)\frac{1}{5+\sqrt{3}}\)

\(=\left(\frac{5+\sqrt{3}}{2}\right).\frac{1}{5+\sqrt{3}}=\frac{1}{2}\)