Question Number 19785 by Tinkutara last updated on 15/Aug/17
$$\mathrm{If}\:{x}^{\left({x}^{\mathrm{4}} \right)} \:=\:\mathrm{4},\:\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of} \\ $$$${x}^{\left({x}^{\mathrm{2}} \right)} \:+\:{x}^{\left({x}^{\mathrm{8}} \right)} ? \\ $$
Answered by Tinkutara last updated on 16/Aug/17
$$\mathrm{Let}\:{x}\:=\:{y}^{\frac{\mathrm{1}}{\mathrm{4}}} \:\Rightarrow\:{x}^{\mathrm{4}} \:=\:{y} \\ $$$${y}^{\frac{{y}}{\mathrm{4}}} \:=\:\mathrm{4} \\ $$$${y}^{{y}} \:=\:\mathrm{4}^{\mathrm{4}} \:\Rightarrow\:{y}\:=\:\mathrm{4}\:\Rightarrow\:{x}\:=\:\sqrt{\mathrm{2}} \\ $$
Commented by ajfour last updated on 16/Aug/17
$$\mathrm{very}\:\mathrm{nice}\:\mathrm{trick}. \\ $$