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Question Number 118171 by mathdave last updated on 15/Oct/20

Commented by Dwaipayan Shikari last updated on 15/Oct/20

$$\sqrt{2}$$

Commented by Dwaipayan Shikari last updated on 15/Oct/20

$$\left(\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{3}}}}\right)\left(\sqrt{4-2\sqrt{2+\sqrt{2+\sqrt{3}}}}\right)$$$$=\sqrt{2}\left(\sqrt{4-2-\sqrt{2+\sqrt{3}}}\right)=\sqrt{2}\left(\sqrt{2-\sqrt{2+\sqrt{3}}}\right)$$$$So$$$$\sqrt{2}\left(\sqrt{2+\sqrt{2+\sqrt{3}}}\right)\times \left(\sqrt{2-\sqrt{2+\sqrt{3}}}\right)$$$$=\sqrt{2}\left(\sqrt{4-2-\sqrt{3}}\right)=\sqrt{2}\left(\sqrt{2-\sqrt{3}}\right)$$$$So$$$$\left(\sqrt{2+\sqrt{3}}\right)\times (\sqrt{2}\left(\sqrt{2-\sqrt{3}}\right)=\sqrt{2}\left(\sqrt{4-3}\right)=\sqrt{2}$$

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