Question

6Use DeMolvre's Theorem to find the power of the complex number and write your solution in the form, a. b.128166432

Answer 1

Given

$a+ib=1+\sqrt[]{3}i$$\begin{gathered} r=\sqrt[]{a^2+b^2} \\ =\sqrt[]{1^2+(-\sqrt[]{3})^2} \\ =\sqrt[]{1+3} \\ =2 \end{gathered}$$\begin{gathered} \theta=\tan ^{-1}\frac{b}{a} \\ \theta=\tan ^{-1}\frac{-\sqrt[]{3}}{1} \\ =-\frac{\pi}{3} \end{gathered}$$\begin{gathered} 1-3i=r(\cos \theta+i\sin \theta) \\ =2(\cos (-\frac{\pi}{3})+i\sin (-\frac{\pi}{3})) \end{gathered}$

Use Demoivre's theorem,

$\begin{gathered} (1-\sqrt[]{3}i)^6=2^6(\cos (-\frac{\pi}{3}\cdot6)+i\sin (-\frac{\pi}{3}\cdot6)) \\ =64(1+i\cdot0) \\ =64 \end{gathered}$

Thus the required solution is 64