# How to find ALL roots of this complex number?

Image from book: https://postimg.cc/VrDBBFdH

Having trouble finding the ladt 2 roots of this complex number for part a). I managed to get z = √2+√6i and -√2-√6i when using k= 0 and 1 with θ = 2pi/3 and -2pi/3 (from 2pi - 8pi/3),

but I cant get the remaining √6+√2i and -√6-√2i with k = 2 and 3, as the theta values give me the same as the previous 2 values of z.
(edited 2 weeks ago)
Original post by MonoAno555
Image from book: https://postimg.cc/VrDBBFdH

Having trouble finding the ladt 2 roots of this complex number for part a). I managed to get z = √2+√6i and -√2-√6i when using k= 0 and 1 with θ = 2pi/3 and -2pi/3 (from 2pi - 8pi/3),

but I cant get the remaining √6+√2i and -√6-√2i with k = 2 and 3, as the theta values give me the same as the previous 2 values of z.

You should know the 4 roots are just a rotation (and scaling) of the usual (4th) roots of unity so once you have the principle one, you could pretty much write the others down as theyre at 90 degrees to each other.

The |z| is 2sqrt(2) and 2pi/3 / 4 = pi/6 which gives one of your missing roots, so it would help to see what you actually wrote down. I think there is a missing - on your first calculated root (Re part) and the second isnt correct (again negative signs).

A bit of thought about the usual side ratios for a 30-60-90 and a sketch with the 4th roots of "unity" should mean you could pretty much write the answers down with little calculation.
(edited 2 weeks ago)