Find an expression for $(\frac{z_1}{z_2})^6$ in terms of $a$ and $i$.

Question

$$ z_1 = 2a\, \text{cis}\frac{\pi}{3}$$

$$z_2 = \sqrt{2}\, \text{cis}\frac{-\pi}{4}$$

What I did: [64a^6 cis(2pi)] / [8 cis(-3pi/2)]

= -8a^6*i

This is the first (or second) time I'm asking a question on stack exchange please don't judge.

Answer 1

Yup! Your answer is $8a^6\exp(\frac{3\pi i}{2})=-8ia^6$. Good work.

*One other method that would've worked is $\displaystyle \left(\frac{z_1}{z_2}\right)=\left(a\sqrt{2}\exp \left(\frac{7\pi i }{12}\right)\right)^6=8a^6\exp\left(\frac{7\pi i}{2}\right)=8a^6\exp\left(\frac{-\pi i}{2}\right)=-8ia^6$

*$\text{cis}(\theta)=\exp(i\theta)$