Question

Solve for the roots in the equation below. In your final answer, include each of the necessary steps and calculations.

x^3 - 27i = 0

Answer 1

`x^3-27i=0`

`x^3=27i`

`x^3=27e^(i\pi/2)`

`x=\left(27e^(i\pi/2)\right)^(1/3)`

`x=3e^(i(\pi/2+2\pi k)/3)`

`x=3e^(i\pi(4k+1)/6)`

where
`k\in\{0,1,2\}`. This means you have


`x=3e^(i\pi/6)=\frac32(\sqrt3+i)`

`x=3e^(i5\pi/6)=\frac32(-\sqrt3+i)`

`x=3e^(i9\pi/6)=-3i`

as the solutions to the original equation.