$(\frac{i+1}{\sqrt{2}})$ in polar coordinates is $\cos(\frac{\pi}{4}) + i\sin(\frac{\pi}{4})$
The roots of this equation are equally spaced on the unit circle around the origin, and the polar angle of $(\frac{i+1}{\sqrt{2}})^4$ is $\pi$
Therefore we know that $(\frac{i+1}{\sqrt{2}})^4 = \cos(\pi) + i\sin(\pi) = -1$
So finally, $\exp[\pi(\frac{i+1}{\sqrt{2}})^4] = \exp[-\pi]$
