Question Number 167107 by mathlove last updated on 06/Mar/22
Answered by alephzero last updated on 06/Mar/22
$${a}^{{b}^{{x}} } \:=\:{a}^{\sqrt[{\mathrm{3}}]{\frac{{b}}{{b}^{{x}} }}} \\ $$$$\Rightarrow\:{b}^{{x}} \:=\:\sqrt[{\mathrm{3}}]{\frac{{b}}{{b}^{{x}} }} \\ $$$$\Rightarrow\:{b}^{{x}} \:=\:\sqrt[{\mathrm{3}}]{{b}^{\mathrm{1}−{x}} } \\ $$$$\Rightarrow\:{b}^{{x}} \:=\:{b}^{\frac{\mathrm{1}−{x}}{\mathrm{3}}} \\ $$$$\Rightarrow\:{x}\:=\:\frac{\mathrm{1}−{x}}{\mathrm{3}} \\ $$$$\Rightarrow\:\mathrm{3}{x}\:=\:\mathrm{1}−{x} \\ $$$$\Rightarrow\:\mathrm{4}{x}\:=\:\mathrm{1} \\ $$$$\Rightarrow\:{x}\:=\:\frac{\mathrm{1}}{\mathrm{4}} \\ $$
Commented by peter frank last updated on 06/Mar/22
$$\mathrm{great} \\ $$