Question Number 194389 by mathlove last updated on 05/Jul/23
$$\sqrt{\sqrt{\mathrm{49}+\mathrm{20}\sqrt{\mathrm{6}}}}=? \\ $$
Answered by som(math1967) last updated on 05/Jul/23
$$\sqrt{\sqrt{\mathrm{5}^{\mathrm{2}} +\left(\mathrm{2}\sqrt{\mathrm{6}}\right)^{\mathrm{2}} +\mathrm{2}.\mathrm{5}.\mathrm{2}\sqrt{\mathrm{6}}}} \\ $$$$\sqrt{\sqrt{\left(\mathrm{5}+\mathrm{2}\sqrt{\mathrm{6}}\right)^{\mathrm{2}} }} \\ $$$$=\sqrt{\mathrm{5}+\mathrm{2}\sqrt{\mathrm{6}}} \\ $$$$=\sqrt{\left(\sqrt{\mathrm{3}}+\sqrt{\mathrm{2}}\right)^{\mathrm{2}} }=\sqrt{\mathrm{3}}+\sqrt{\mathrm{2}} \\ $$