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

Use Demoivre's theorem,

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

Thus the required solution is 64