Question

Use the polar representation to write the following in the form a + bi (a) (-1+ i)^7 (b) (1+v3i)^10

Answer 1

Answer with Step-by-step explanation:

For any complex number x+iy the polar form is represented as

$z=re^{i\theta }$

where

$r^{2}=x^2+y^2\\\\tan(\theta )=\frac{y}{x}$

Part a) x+iy = -1+i

Thus $r=\sqrt{-1^2+1^2}=\sqrt{2}\\\\ \theta _{1}=tan^{-1}(\frac{-1}{1})=\frac{-\pi }{4}$

Thus using De-Morvier's theorem

$(-1+i)^7=(\sqrt{2}e^{i\frac{-\pi }{4}})^7\\\\\therefore (-1+i)^7=2^{7/2}\cdot e^{\frac{i-\pi }{28}}$

Part 2) x+iy = 1+3i

Thus $r=\sqrt{1^2+3^2}=\sqrt{10}\\\\ \theta _{1}=tan^{-1}(\frac{3}{1})=1.25$

Thus using De-Morvier's theorem

$(1+3i)^{10}=(\sqrt{10}e^{1.25i})^{10}\\\\\therefore (1+3i)^{10}=10^{5}\cdot e^{12.5i}$