Put the following in rectangular form.

Question

$$(\sqrt{3}+i)^7$$

My question: $r = 2$. For $\theta$, do I use $\dfrac{\pi}{6}$ or $\dfrac{\pi}{6} + 2n\pi$? The book uses the former but I thought the latter is more appropriate.

Thank you.

Answer 1

$$(\sqrt{3}+i)^7=2^7(\frac{\sqrt{3}}{2}+\frac{1}{2}i)^7=2^7e^{7i\pi/6}=2^7(\cos\frac{7\pi}{6}+i\sin\frac{7\pi}{6})$$