Question Number 153682 by EDWIN88 last updated on 09/Sep/21
$$\:\left(\sqrt[{{i}}]{{i}}\:\right)^{{xi}} \:=\:{i}^{{x}} \: \\ $$$$\:\:{x}=?\: \\ $$
Answered by MJS_new last updated on 09/Sep/21
$$\mathrm{lhs}\:\mathrm{i}=\mathrm{e}^{\mathrm{i}\frac{\pi}{\mathrm{2}}} \:\Rightarrow\:\sqrt[{\mathrm{i}}]{\mathrm{i}}=\mathrm{e}^{\frac{\pi}{\mathrm{2}}} \:\Rightarrow\:\left(\sqrt[{\mathrm{i}}]{\mathrm{i}}\right)^{\mathrm{i}{x}} =\mathrm{e}^{\mathrm{i}\frac{{x}\pi}{\mathrm{2}}} \\ $$$$\mathrm{rhs}\:\mathrm{i}^{{x}} =\mathrm{e}^{\mathrm{i}\frac{{x}\pi}{\mathrm{2}}} \\ $$$$\Rightarrow\:{x}\in\mathbb{C} \\ $$