Question Number 161884 by HongKing last updated on 23/Dec/21
$$\mathrm{Solve}\:\mathrm{for}\:\mathrm{real}\:\mathrm{numbers}: \\ $$$$\sqrt[{\mathrm{7}}]{\mathrm{2}^{\boldsymbol{\mathrm{x}}} \:+\:\mathrm{2}^{-\mathrm{1}} }\:=\:\mathrm{1}\:+\:\sqrt[{\mathrm{7}}]{\mathrm{2}^{\boldsymbol{\mathrm{x}}} \:-\:\mathrm{2}^{-\mathrm{1}} } \\ $$
Commented by mr W last updated on 23/Dec/21
$$\sqrt[{\mathrm{7}}]{\mathrm{2}^{{x}} +\frac{\mathrm{1}}{\mathrm{2}}}=\mathrm{1}+\sqrt[{\mathrm{7}}]{\mathrm{2}^{{x}} −\frac{\mathrm{1}}{\mathrm{2}}} \\ $$$$\mathrm{1}=\mathrm{1}+\mathrm{0}\:? \\ $$$$\Rightarrow\mathrm{2}^{{x}} =\frac{\mathrm{1}}{\mathrm{2}}\:\checkmark \\ $$$$\Rightarrow{x}=−\mathrm{1} \\ $$
Commented by HongKing last updated on 25/Dec/21
$$\mathrm{thank}\:\mathrm{you}\:\mathrm{my}\:\mathrm{dear}\:\mathrm{Sir} \\ $$