Question Number 182247 by Shrinava last updated on 06/Dec/22
$$\mathrm{Find}: \\ $$$$\sqrt{\mathrm{9}\:+\:\mathrm{4}\:\sqrt{\mathrm{5}}}\:−\:\sqrt{\mathrm{9}\:−\:\mathrm{4}\:\sqrt{\mathrm{5}}}\:=\:? \\ $$
Answered by mr W last updated on 06/Dec/22
$$\mathrm{9}\pm\mathrm{4}\sqrt{\mathrm{5}}=\left(\sqrt{\mathrm{5}}\right)^{\mathrm{2}} \pm\mathrm{4}\sqrt{\mathrm{5}}+\mathrm{2}^{\mathrm{2}} =\left(\sqrt{\mathrm{5}}\pm\mathrm{2}\right)^{\mathrm{2}} \\ $$$$\sqrt{\mathrm{9}+\mathrm{4}\sqrt{\mathrm{5}}}−\sqrt{\mathrm{9}−\mathrm{4}\sqrt{\mathrm{5}}}=\sqrt{\mathrm{5}}+\mathrm{2}−\left(\sqrt{\mathrm{5}}−\mathrm{2}\right)=\mathrm{4} \\ $$
Answered by mr W last updated on 06/Dec/22
$$\left(\sqrt{\mathrm{9}\:+\:\mathrm{4}\:\sqrt{\mathrm{5}}}\:−\:\sqrt{\mathrm{9}\:−\:\mathrm{4}\:\sqrt{\mathrm{5}}}\right)^{\mathrm{2}} \:=\mathrm{18}−\mathrm{2}=\mathrm{16} \\ $$$$\Rightarrow\sqrt{\mathrm{9}\:+\:\mathrm{4}\:\sqrt{\mathrm{5}}}\:−\:\sqrt{\mathrm{9}\:−\:\mathrm{4}\:\sqrt{\mathrm{5}}}\:=\sqrt{\mathrm{16}}=\mathrm{4} \\ $$
Commented by Shrinava last updated on 07/Dec/22
$$\mathrm{thank}\:\mathrm{you}\:\mathrm{dear}\:\mathrm{professor} \\ $$