Question Number 167966 by mathlove last updated on 30/Mar/22
Answered by HeferH last updated on 30/Mar/22
$$\:{x}^{{x}^{\mathrm{4}} } \:=\:\mathrm{4} \\ $$$$\:\left({x}^{{x}^{\mathrm{4}} } \right)^{\mathrm{4}} \:=\:\left(\mathrm{4}\right)^{\mathrm{4}} \\ $$$$\:\left({x}^{\mathrm{4}} \right)^{{x}^{\mathrm{4}} } \:=\:\mathrm{4}^{\mathrm{4}} \\ $$$$\:{x}^{\mathrm{4}} \:=\:\mathrm{4} \\ $$$$\:{x}\:=\:\sqrt[{\mathrm{4}}]{\mathrm{4}} \\ $$$$\:{x}\:=\:\left(\mathrm{2}^{\mathrm{2}} \right)^{\frac{\mathrm{1}}{\mathrm{4}}} \:\rightarrow\:\mathrm{2}^{\frac{\mathrm{1}}{\mathrm{2}}} \\ $$$$\:{x}\:=\:\sqrt{\mathrm{2}} \\ $$$$\:\left(\sqrt{\mathrm{2}}\right)^{\left(\sqrt{\mathrm{2}}\right)^{\mathrm{8}} } \:+\:\left(\sqrt{\mathrm{2}}\right)^{\left(\sqrt{\mathrm{2}}\right)^{\mathrm{2}} } \:=\:\:\left(\mathrm{2}^{\frac{\mathrm{1}}{\mathrm{2}}} \right)^{\mathrm{16}} \:+\:\left(\mathrm{2}^{\frac{\mathrm{1}}{\mathrm{2}}} \right)^{\mathrm{2}} \\ $$$$\:\:\:\:\left(\mathrm{2}^{\frac{\mathrm{1}}{\mathrm{2}}} \right)^{\mathrm{16}} \:+\:\left(\mathrm{2}^{\frac{\mathrm{1}}{\mathrm{2}}} \right)^{\mathrm{2}} \:\:\:\:\:\:\:=\:\:\mathrm{2}^{\mathrm{8}} \:+\:\mathrm{2}\:=\:\mathrm{258} \\ $$$$\: \\ $$