Question

Use DeMoivre's Theorem to find the power of the complex number and write your solution in the form, a + bi.(1 - 15 = ?O41o-4-41

Answer 1

`(1-i)^5`

Demoiver's Theorem

`\begin{gathered} (r(\cos \theta+i\sin \theta))^n=r^n\cdot(\cos n\theta+i\sin n\theta) \\ r=\sqrt[]{1^2+1^2}=\sqrt[]{2} \\ \end{gathered}`

(1 - i) it is u=in the fourth quadrant and it is equal to 315 degrees

`\begin{gathered} (\sqrt[]{2})^5\cdot(cos(5\cdot315)+i\cdot\sin (5\cdot315)) \\ 4\cdot(\sqrt[]{2})^5\cdot(\cos 135+i\sin 135) \\ 4\cdot(\sqrt[]{2})^5\cdot(-\frac{\sqrt[]{2}}{2}+i\frac{\sqrt[]{2}}{2}) \\ -4\cdot(-\frac{(\sqrt[]{2})^6}{2}+i\frac{(\sqrt[]{2})^6}{2}) \\ -4+4i \end{gathered}`