Get to the root of the problem

Calculus Level 4

\[\large \begin{cases} f(x)=\sqrt{\frac{x}{\sqrt[3]{\frac{x}{\sqrt[4]{\frac{x}{\ddots}}}}}} \\ g(x)=\sqrt{x\sqrt[3]{x\sqrt[4]{x\sqrt[5]{\cdots}}}} \end{cases} \]

The functions \(f(x)\) and \(g(x)\) are defined for positive real numbers \(x\) as above. What is the value of \(y\) for which \(f(y)g(y)=2016\)?

Give your answer as \(\lfloor y \rfloor\).

Notation: \( \lfloor \cdot \rfloor \) denotes the floor function.

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