# Imaginary number challenge (3)

**Algebra**Level 4

\[\large\left(\left(\left(\left(i^{2i}\right)^{3i}\right)^{4i}\right)^{\cdots}\right)^{2016i}\]

If the *principal value* of the expression above can be written in the form \(\exp\left(\dfrac{x!\pi}y\right)\) where \(x\) and \(y\) are positive integers, find \(x+y\).

**Details and Assumptions:**

- Every complex number \(z\) can be written in the form \(z=re^{i\theta+2k\pi}\) for some real \(r,\theta\) and all \(k\) such that \(-\pi\lt \theta\leq\pi\). The value when \(k=0\) is called the
*principal value*of \(z\). - \(i=\sqrt{-1}\) denotes the imaginary unit.
- \(\exp(x)=e^x\) where \(e\) denotes the Euler's number.
- \(n!\) denotes the factorial function. For example, \(8! = 1\times2\times3\times\cdots\times8 \).