# Discrete signal testing for periodicity

How would one go about determining if the following discrete time signal x[n] is periodic, and if it is, determine its fundamental period?

I understand that the period for the second exponential term is 6, but apart from that I am unable to further my calculation.

The answer to the question above is:

but I fail to understand how the equation highlighted in yellow comes to be. Any help with this matter would be appreciated.

it is more clear if you utilize the Euler's formula \begin{align} x[n+N] &= e^{-2}e^{j(\frac{\pi}{3}(n+N)-\frac{\pi}{4})} \\ &= e^{-2}\Big[ \cos(\frac{\pi}{3}n+\frac{\pi}{3}N-\frac{\pi}{4}) -j \sin(\frac{\pi}{3}n+\frac{\pi}{3}N-\frac{\pi}{4}) \Big] \end{align} Using trigonometric identities ( I will leave it as an exercise to you), $$x[n]=x[n+N]$$ holds if N=6 which is the smallest value that makes the equality to hold. Indeed, the signal is periodic with a fundamental period $$N_o=6$$.