Question

Find the value of the nested square roots

\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}}}}

Collected in the board: Square Root

Steven Zheng posted 49 minutes ago

Answer

Repeating using of difference of squares formula

\begin{alignedat}{2} &\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{\bigg(2+\sqrt{2+\sqrt{2+\sqrt{3}}}\bigg)\bigg(2-\sqrt{2+\sqrt{2+\sqrt{3}}}\bigg)} \\ &=\sqrt{2+\sqrt{3}}\cdot\sqrt{2+\sqrt{2+\sqrt{3}}}\cdot\sqrt{2-\sqrt{2+\sqrt{3}}} \\ &=\sqrt{2+\sqrt{3}}\cdot\Big(\sqrt{2-\sqrt{3}} \Big) \\ &=1 \end{alignedat}

Steven Zheng posted 48 minutes ago

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