Complex exponential factorials!

$\begin{aligned} \LARGE \sqrt[{\left(x × \sqrt[4]{x^3}\right)}^{x!}]{x!} & = \LARGE {\left(\sqrt[{\left(i^2\right)}^{x!}]{x!}\right)}^{x^{-8820}}\\ \\ \LARGE \sqrt[x^{\frac{7}{4}x!}]{x!} & = \LARGE {\left(\sqrt[{-1}^{x!}]{x!}\right)}^{x^{-8820}}\\ \\ \LARGE {x!}^{\tfrac{1}{x^{\frac{7}{4}x!}}} & = \LARGE {x!}^{x^{-8820}}\\ \\ \LARGE {x!}^{x^{-\frac{7}{4}x!}} & = \LARGE {x!}^{x^{-8820}}\\ \\ \text{Equating the powers, we get}:-\\ \LARGE x^{-\tfrac{7}{4}x!} & = \LARGE x^{-8820}\\ \\ \text{Equating the powers, we get}:-\\ \Large {\cancel{-}\dfrac{7}{4}x!} & = \Large {\cancel{-}8820}\\ \\ \Large x! & = \Large \dfrac{8820 × 4}{7}\\ \\ \Large x! & = \Large 5040\\ \\ \Large x! & = \Large 7×6×5×4×3×2×1\\ \\ \Large x & = \Large \boxed{7} \end{aligned}$

In step 2, $i^{x!} = 1$ , because x! is even for all $x \not\in \{-1 , 0 , 1\}$ because they are multiple of $2$ .