Ramanujan's nested radical is great! (4)

Calculus Level 5

Find the value of \(x\) satisfying the equation below.

\[\large \dfrac{(\sqrt[0!]{2016^{\pi^{0}}})(\sqrt[1!]{2016^{\pi^{1}}})(\sqrt[4!]{2016^{\pi^{4}}})(\sqrt[5!]{2016^{\pi^{5}}})(\sqrt[8!]{2016^{\pi^{8}}})(\sqrt[9!]{2016^{\pi^{9}}})(\dots)}{(\sqrt[2!]{2016^{\pi^{2}}})(\sqrt[3!]{2016^{\pi^{3}}})(\sqrt[6!]{2016^{\pi^{6}}})(\sqrt[7!]{2016^{\pi^{7}}})(\sqrt[10!]{2016^{\pi^{10}}})(\sqrt[11!]{2016^{\pi^{11}}})(\dots)}=2016^{x}\]

Notation : \(n!\) denotes the factorial function. For example, \(8! = 1\times2\times3\times\cdots\times8 \).

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