# 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, $$8! = 1\times2\times3\times\cdots\times8$$.
×