The easiest method is to use De Moivre's theorem. For complex number #z#
#z= r(costheta + isintheta)#
#z^n = r^n(cosntheta + isinntheta)#
So we want to convert our complex number to polar form. The modulus #r# of a complex number #a+bi# is given by
#r = sqrt(a^2+b^2)#
#r = sqrt((1/2)^2 + (sqrt(3)/2)^2) = sqrt(1/4 + 3/4) = 1#
The complex number will be in the first quadrant of an Argand diagram so the argument is given by:
#theta = tan^(-1)(b/a)#
#theta = tan^(-1)((sqrt(3)/2)/(1/2)) = tan^(-1)(sqrt(3)) = pi/3#
#z = cos(pi/3) + isin(pi/3)#
#z^2 = cos(2pi/3) + isin(2pi/3) = 1/2(-1 + sqrt(3)i)#
#z^3 = cos(3pi/3) + isin(3pi/3) = -1#
#z^4 = cos(4pi/3) + isin(4pi/3) = -1/2(1+sqrt(3)i)#
