\(\sqrt{\dfrac{2\sqrt{10}+\sqrt{30}-2\sqrt{2}-\sqrt{6}}{2\sqrt{10}-2\sqrt{2}}}:\dfrac{2}{\sqrt{3}-1}\)
\(=\sqrt{\dfrac{\sqrt{10}\left(2+\sqrt{3}\right)-\sqrt{2}\left(2+\sqrt{3}\right)}{2\sqrt{10}-2\sqrt{2}}}:\dfrac{2}{\sqrt{3}-1}\)
\(=\sqrt{\dfrac{\left(2+\sqrt{3}\right)\left(\sqrt{10}-\sqrt{2}\right)}{2\left(\sqrt{10}-\sqrt{2}\right)}}\cdot\dfrac{\sqrt{3}-1}{2}\)
\(=\sqrt{\dfrac{2+\sqrt{3}}{2}}\cdot\dfrac{\sqrt{3}-1}{2}\)
\(=\sqrt{\dfrac{2\cdot\left(2+\sqrt{3}\right)}{2\cdot2}}\cdot\dfrac{\sqrt{3}-1}{2}\)
\(=\sqrt{\dfrac{4+2\sqrt{3}}{4}}\cdot\dfrac{\sqrt{3}-1}{2}\)
\(=\dfrac{\sqrt{\left(\sqrt{3}\right)^2+2\sqrt{3}\cdot1+1^2}}{\sqrt{4}}\cdot\dfrac{\sqrt{3}-1}{2}\)
\(=\dfrac{\sqrt{\left(\sqrt{3}+1\right)^2}}{2}\cdot\dfrac{\sqrt{3}-1}{2}\)
\(=\dfrac{\left|\sqrt{3}+1\right|}{2}\cdot\dfrac{\sqrt{3}-1}{2}\)
\(=\dfrac{\left(\sqrt{3}+1\right)\left(\sqrt{3}-1\right)}{2\cdot2}\)
\(=\dfrac{3-1}{4}\)
\(=\dfrac{2}{4}\)
\(=\dfrac{1}{2}\)
