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Question Number 153682 by EDWIN88 last updated on 09/Sep/21

 ((i)^(1/i)  )^(xi)  = i^x      x=?

$$\:\left(\sqrt[{{i}}]{{i}}\:\right)^{{xi}} \:=\:{i}^{{x}} \: \\ $$$$\:\:{x}=?\: \\ $$

Answered by MJS_new last updated on 09/Sep/21

lhs i=e^(i(π/2))  ⇒ (i)^(1/i) =e^(π/2)  ⇒ ((i)^(1/i) )^(ix) =e^(i((xπ)/2))   rhs i^x =e^(i((xπ)/2))   ⇒ x∈C

$$\mathrm{lhs}\:\mathrm{i}=\mathrm{e}^{\mathrm{i}\frac{\pi}{\mathrm{2}}} \:\Rightarrow\:\sqrt[{\mathrm{i}}]{\mathrm{i}}=\mathrm{e}^{\frac{\pi}{\mathrm{2}}} \:\Rightarrow\:\left(\sqrt[{\mathrm{i}}]{\mathrm{i}}\right)^{\mathrm{i}{x}} =\mathrm{e}^{\mathrm{i}\frac{{x}\pi}{\mathrm{2}}} \\ $$$$\mathrm{rhs}\:\mathrm{i}^{{x}} =\mathrm{e}^{\mathrm{i}\frac{{x}\pi}{\mathrm{2}}} \\ $$$$\Rightarrow\:{x}\in\mathbb{C} \\ $$

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