What Are The Fourth Roots Of 3 3 3i

Solution

You can do this directly since there is an expression for the square root of a complex number in Cartesian form, but it is easier to convert to polar form and use De Moivre’s theorem.

The modulus of the number is 6 = sqrt(9 + 27) and the principle argument is 2π/3 (120°) which you get from plotting the point in polar co-ordinates and noting that √3=tan 60°.

Then take the 4th root of the modulus and divide the argument by 4 to get your answers (there are 4 of them equally spaced around the circle defined by the root of the modulus).