Answer with Step-by-step explanation:
For any complex number x+iy the polar form is represented as
[tex]z=re^{i\theta }[/tex]
where
[tex]r^{2}=x^2+y^2\\\\tan(\theta )=\frac{y}{x}[/tex]
Part a) x+iy = -1+i
Thus [tex]r=\sqrt{-1^2+1^2}=\sqrt{2}\\\\ \theta _{1}=tan^{-1}(\frac{-1}{1})=\frac{-\pi }{4}[/tex]
Thus using De-Morvier's theorem
[tex](-1+i)^7=(\sqrt{2}e^{i\frac{-\pi }{4}})^7\\\\\therefore (-1+i)^7=2^{7/2}\cdot e^{\frac{i-\pi }{28}}[/tex]
Part 2) x+iy = 1+3i
Thus [tex]r=\sqrt{1^2+3^2}=\sqrt{10}\\\\ \theta _{1}=tan^{-1}(\frac{3}{1})=1.25[/tex]
Thus using De-Morvier's theorem
[tex](1+3i)^{10}=(\sqrt{10}e^{1.25i})^{10}\\\\\therefore (1+3i)^{10}=10^{5}\cdot e^{12.5i}[/tex]
