Question

How do you simplify `[(3+2i)^ 3 / (-2+3i)^4]`?

Answer 1

`(3-2i)/13`

Explanation:

`3+2i=sqrt(3^2+2^2)e^{i phi}`
`-2+3i=sqrt(3^2+2^2)e^{i( phi+pi/2)}`

with `phi = arctan(2/3)`

`[(3+2i)^ 3 / (-2+3i)^4] =((3+2i)/(-2+3i))^3/( (-2+3i))=e^{-i (3pi)/2}/(sqrt(3^2+2^2)e^{i( phi+pi/2)})`
`=1/sqrt(3^2+2^2)e^{-i(phi+2pi)} = 1/sqrt(3^2+2^2)e^{-i phi} =`
`sqrt(3^2+2^2)/(3^2+2^2)e^{-i phi} = (3-2i)/13`