Question

\(\left(\sqrt{3+\sqrt{15}-\sqrt{3-\sqrt{5}}}\right)^2\)

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Answer 1

\(\left(\sqrt{3+\sqrt{15}-\sqrt{3-\sqrt{5}}}\right)^2=3+\sqrt{15}-\sqrt{3-\sqrt{5}}=\dfrac{\sqrt{2}\left(3+\sqrt{15}-\sqrt{3-\sqrt{5}}\right)}{\sqrt{2}}=\dfrac{3\sqrt{2}+\sqrt{30}-\sqrt{6-2\sqrt{5}}}{\sqrt{2}}=\dfrac{3\sqrt{2}+\sqrt{30}-\sqrt{5-2\sqrt{5}+1}}{\sqrt{2}}=\dfrac{3\sqrt{2}+\sqrt{30}-\sqrt{\left(\sqrt{5}-1\right)^2}}{\sqrt{2}}=\dfrac{3\sqrt{2}+\sqrt{30}-\left|\sqrt{5}-1\right|}{\sqrt{2}}=\dfrac{3\sqrt{2}+\sqrt{30}-\sqrt{5}+1}{\sqrt{2}}=\dfrac{\sqrt{2}\left(3\sqrt{2}+\sqrt{30}-\sqrt{5}+1\right)}{2}=\dfrac{6+2\sqrt{15}-\sqrt{10}+\sqrt{2}}{2}\)