# Smallest positive integer N power of a exponential complex number such that it is real.

How would I find the smallest positive integer N such that $(w)^N$ is a real number?

Given that

$$w = 8e^{\dfrac{i7\pi}{6}}.$$

• Hint. Start computing some powers and remember the value of $e^{i\pi}$ – Ethan Bolker Nov 18 '17 at 18:19

So $w = 8e^{i7 {\pi} /6}$. Use an identity which says $e^{i {\theta}} = cos {\theta} + isin {\theta}$. So sin and cos are real valued for reals. So we have to make $sin{N7 {\pi}/6} = 0$. Since $sin {n {\pi}} = 0$ for all n in integers. So make $N7 {\pi}/6$ into an integer and $N \geq 1$ will give $N = 6$.