Question

\(\dfrac{\sqrt{10}-\sqrt{2}}{\sqrt{5}-1}\) - \(\dfrac{7}{\sqrt{7}}\)  + \(\dfrac{5}{\sqrt{7}-\sqrt{2}}\)

\(2\sqrt{27}\) - 6\(\sqrt{\dfrac{1}{3}}\) + \(\dfrac{1}{2+\sqrt{3}}\) - \(\dfrac{9}{\sqrt{3}}\)

Answer 1

\(\dfrac{\sqrt{10}-\sqrt{2}}{\sqrt{5}-1}-\dfrac{7}{\sqrt{7}}+\dfrac{5}{\sqrt{7}-\sqrt{2}}\)

\(=\dfrac{\sqrt{2}\left(\sqrt{5}-1\right)}{\sqrt{5}-1}-\sqrt{7}+5\dfrac{\sqrt{7}+\sqrt{2}}{\left(\sqrt{7}-\sqrt{2}\right)\left(\sqrt{7}+\sqrt{2}\right)}\)

\(=\sqrt{2}-\sqrt{7}+\sqrt{7}+\sqrt{2}=2\sqrt{2}\)

\(2\sqrt{27}-6\sqrt{\dfrac{1}{3}}+\dfrac{1}{2+\sqrt{3}}-\dfrac{9}{\sqrt{3}}\)

\(=2\cdot3\sqrt{3}-2\sqrt{3}+\dfrac{2-\sqrt{3}}{\left(2+\sqrt{3}\right)\left(2-\sqrt{3}\right)}-3\sqrt{3}\)

\(=6\sqrt{3}-2\sqrt{3}-3\sqrt{3}+2-\sqrt{3}=2\)