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 )=(y)/(x)`

Part a) x+iy = -1+i

Thus
`r=√(-1^2+1^2)=√(2)\\\\  \theta _(1)=tan^(-1)((-1)/(1))=(-\pi )/(4)`

Thus using De-Morvier's theorem

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

Part 2) x+iy = 1+3i

Thus
`r=√(1^2+3^2)=√(10)\\\\  \theta _(1)=tan^(-1)((3)/(1))=1.25`

Thus using De-Morvier's theorem

`(1+3i)^(10)=(√(10)e^(1.25i))^(10)\\\\\therefore (1+3i)^(10)=10^(5)\cdot e^(12.5i)`