Question Number 43349 by peter frank last updated on 10/Sep/18
Answered by math1967 last updated on 10/Sep/18
$$\sqrt{\mathrm{2}+\sqrt{\mathrm{5}}−\sqrt{\mathrm{6}−\mathrm{3}\sqrt{\mathrm{5}}+\sqrt{\left(\mathrm{3}−\sqrt{\left.\mathrm{5}\right)^{\mathrm{2}} }\right.}}} \\ $$$$\sqrt{\mathrm{2}+\sqrt{\mathrm{5}}−\sqrt{\mathrm{6}−\mathrm{3}\sqrt{\mathrm{5}}+\mathrm{3}−\sqrt{\mathrm{5}}}} \\ $$$$\sqrt{\mathrm{2}+\sqrt{\mathrm{5}}−\sqrt{\left(\sqrt{\mathrm{5}}−\mathrm{2}\right)^{\mathrm{2}} }} \\ $$$$\sqrt{\mathrm{2}+\sqrt{\mathrm{5}}−\sqrt{\mathrm{5}}+\mathrm{2}} \\ $$$$\sqrt{\mathrm{4}}=\mathrm{2}\:\:\left({proved}\right) \\ $$
Commented by peter frank last updated on 10/Sep/18
$${much}\:{respect} \\ $$
Answered by Joel578 last updated on 10/Sep/18
$$\sqrt{\mathrm{14}\:−\:\mathrm{6}\sqrt{\mathrm{5}}}\:=\:\sqrt{\mathrm{14}\:−\:\mathrm{2}\sqrt{\mathrm{45}}}\:=\:\mathrm{3}\:−\:\sqrt{\mathrm{5}} \\ $$$$ \\ $$$$\sqrt{\mathrm{6}\:−\:\mathrm{3}\sqrt{\mathrm{5}}\:+\:\mathrm{3}\:−\:\sqrt{\mathrm{5}}}\:=\:\sqrt{\mathrm{9}\:−\:\mathrm{4}\sqrt{\mathrm{5}}}\:=\:\sqrt{\mathrm{9}\:−\:\mathrm{2}\sqrt{\mathrm{20}}} \\ $$$$=\:\sqrt{\mathrm{5}}\:−\:\mathrm{2} \\ $$$$ \\ $$$$\sqrt{\mathrm{2}\:+\:\sqrt{\mathrm{5}}\:−\:\sqrt{\mathrm{5}}\:+\:\mathrm{2}}\:=\:\sqrt{\mathrm{4}}\:=\:\mathrm{2} \\ $$
Commented by malwaan last updated on 10/Sep/18
$$\mathrm{3}−\sqrt{\mathrm{5}}\:?? \\ $$
Commented by Joel578 last updated on 10/Sep/18
$$\left(\mathrm{3}\:−\:\sqrt{\mathrm{5}}\right)^{\mathrm{2}} \:=\:\mathrm{9}\:−\:\mathrm{6}\sqrt{\mathrm{5}}\:+\:\mathrm{5}\:=\:\mathrm{14}\:−\:\mathrm{6}\sqrt{\mathrm{5}}\:=\:\mathrm{14}\:−\:\mathrm{2}\sqrt{\mathrm{45}} \\ $$$$\sqrt{\left(\mathrm{3}\:−\:\sqrt{\mathrm{5}}\right)^{\mathrm{2}} }\:=\:\sqrt{\mathrm{14}\:−\:\mathrm{2}\sqrt{\mathrm{45}}} \\ $$
Commented by peter frank last updated on 10/Sep/18
$$ \\ $$$$ \\ $$$${thanks} \\ $$