Question Number 118171 by mathdave last updated on 15/Oct/20
Commented by Dwaipayan Shikari last updated on 15/Oct/20
$$\sqrt{\mathrm{2}} \\ $$
Commented by Dwaipayan Shikari last updated on 15/Oct/20
$$\left(\sqrt{\mathrm{2}+\sqrt{\mathrm{2}+\sqrt{\mathrm{2}+\sqrt{\mathrm{3}}}}}\right)\left(\sqrt{\mathrm{4}−\mathrm{2}\sqrt{\mathrm{2}+\sqrt{\mathrm{2}+\sqrt{\mathrm{3}}}}}\right) \\ $$$$=\sqrt{\mathrm{2}}\left(\sqrt{\mathrm{4}−\mathrm{2}−\sqrt{\mathrm{2}+\sqrt{\mathrm{3}}}}\right)=\sqrt{\mathrm{2}}\left(\sqrt{\mathrm{2}−\sqrt{\mathrm{2}+\sqrt{\mathrm{3}}}}\right) \\ $$$${So} \\ $$$$\sqrt{\mathrm{2}}\left(\sqrt{\mathrm{2}+\sqrt{\mathrm{2}+\sqrt{\mathrm{3}}}}\right)×\left(\sqrt{\mathrm{2}−\sqrt{\mathrm{2}+\sqrt{\mathrm{3}}}}\right) \\ $$$$=\sqrt{\mathrm{2}}\left(\sqrt{\mathrm{4}−\mathrm{2}−\sqrt{\mathrm{3}}}\right)=\sqrt{\mathrm{2}}\left(\sqrt{\mathrm{2}−\sqrt{\mathrm{3}}}\right) \\ $$$${So} \\ $$$$\left(\sqrt{\mathrm{2}+\sqrt{\mathrm{3}}}\right)×\left(\sqrt{\mathrm{2}}\left(\sqrt{\mathrm{2}−\sqrt{\mathrm{3}}}\right)=\sqrt{\mathrm{2}}\left(\sqrt{\mathrm{4}−\mathrm{3}}\right)=\sqrt{\mathrm{2}}\right. \\ $$