Nested roots, varying powers!

$\begin{aligned} \huge \sqrt[10]{\sqrt[20]{\sqrt[30]{{\ddots}_{\sqrt[100]{10}}}}} & = \huge \sqrt[(x - 1)! × x^{x + 1}]{x}\\ \\ \Large {10}^{\tfrac{1}{10} × \tfrac{1}{20} × \tfrac{1}{30} × \cdots × \tfrac{1}{100}} & = \Large x^{\tfrac{1}{(x - 1)! × x^{x + 1}}}\\ \\ \Large {10}^{\tfrac{1}{1 × 2 × 3 × \cdots × 10} × \tfrac{1}{{10}^{10}}} & = \Large x^{\tfrac{1}{(x - 1)! × x × x^x}}\\ \\ \Large {10}^{\tfrac{1}{10! × {10}^{10}}} & = \Large x^{\tfrac{1}{x! × x^x}}\\ \\ \Large \therefore x & = \boxed{10} \end{aligned}$