Question

How do you find the power `(sqrt3-i)^3` and express the result in rectangular form?

Answer 1

`(sqrt(3)-i)^3 = -8i`

Explanation:

Let `z=sqrt(3)-i`

First let us plot the point `z` on the Argand diagram:

And we will put the complex number into polar form:

`|z| = sqrt(3+1) = 2`
`arg(z) = tan^-1(-1/sqrt(3)) = -(pi)/6`

So then in polar form we have:

`z = 2(cos(-(pi)/6) + isin(-(pi)/6))`

By De Moivre's Theorem:

`z^3 = {2(cos(-(pi)/6) + isin(-(pi)/6))}^3`
`\ \ \ = 2^3(cos(-(3pi)/6) + isin(-(3pi)/6))`
`\ \ \ = 8(cos(-(pi)/2) + isin(-(pi)/2))`
`\ \ \ = 8(0-i)`
`\ \ \ = -8i`