Exponentiation (including fractional exponentiation, aka roots) of complex numbers is easily done when the number is converted first to polar form.
If we graph the given complex number in the complex plane (x-axis is real, y-axis imaginary), we can see that it lies in Q IV, 9 units away from the origin (0 + 0i), at an angle of 11π/6. Thus, the given complex number can be written as 9(cos11π/6 + isin11π/6), also sometimes written as 9cis11π/6 or as 9e11π/6i.
To get its primary 4th root, we take the fourth root of its radius, 9, and divide its angle by 4. So we get √3(cos11π/24 + isin11π/24). The remaining 3 roots also have a radius = √3, and are distributed in the complex plane with angles that differ by 2π/4 = π/2. This gives us √3cis23π/24 , √3cis35π/24 ,and √3cis47π/24.
To verify our answers, we could convert any or all of these back to rectangular form and raise them to the fourth power by hand, using binomial expansion.
