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

Given that

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

  • 1
    $\begingroup$ Hint. Start computing some powers and remember the value of $e^{i\pi}$ $\endgroup$ – 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 $N = 6$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.