Question

\(R=\sqrt{2+\sqrt{3}}\times\sqrt{2+\sqrt{2+\sqrt{3}}}\times\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{3}}}}\times\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{3}}}}\)

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Answer 1

\(R=\sqrt{2+\sqrt{3}}\cdot\sqrt{2+\sqrt{2+\sqrt{3}}}\cdot\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{3}}}}\cdot\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{3}}}}\\ =\sqrt{2+\sqrt{3}}\cdot\sqrt{2+\sqrt{2+\sqrt{3}}}\cdot\sqrt{4-2-\sqrt{2+\sqrt{3}}}\\ =\sqrt{2+\sqrt{3}}\cdot\sqrt{2+\sqrt{2+\sqrt{3}}}\cdot\sqrt{2-\sqrt{2+\sqrt{3}}}\\ =\sqrt{2+\sqrt{3}}\cdot\sqrt{4-2-\sqrt{3}}\\ =\sqrt{2+\sqrt{3}}\cdot\sqrt{2-\sqrt{3}}\\ =\sqrt{4-3}=1\)