Question Number 172262 by Mikenice last updated on 25/Jun/22
Answered by Rasheed.Sindhi last updated on 25/Jun/22
$$\sqrt[{\mathrm{3}}]{\mathrm{3}\sqrt[{\mathrm{3}}]{\mathrm{3}\sqrt[{\mathrm{3}}]{\mathrm{3}…}}}\:=\sqrt{{n}}\:>\mathrm{0} \\ $$$$\left(\sqrt[{\mathrm{3}}]{\mathrm{3}\sqrt[{\mathrm{3}}]{\mathrm{3}\sqrt[{\mathrm{3}}]{\mathrm{3}…}}}\:\right)^{\mathrm{3}} =\left(\sqrt{{n}}\:\right)^{\mathrm{3}} \\ $$$$\mathrm{3}\sqrt[{\mathrm{3}}]{\mathrm{3}\sqrt[{\mathrm{3}}]{\mathrm{3}…}}\:=\left(\sqrt{{n}}\:\right)^{\mathrm{3}} \\ $$$$\mathrm{3}\sqrt{{n}}=\left(\sqrt{{n}}\:\right)^{\mathrm{3}} \\ $$$$\mathrm{3}=\left(\sqrt{{n}}\:\right)^{\mathrm{2}} \:;\:{n}\neq\mathrm{0} \\ $$$${n}=\mathrm{3} \\ $$