Find the four complex roots of the equation z^4 = 8(3^0.5+i) in the form z = re^(i*theta)

We know that z=re^(itheta) from the definition of the exponential form of a complex number. Hence it follows that: z^4=(re^(itheta))^4=r^4e^(4itheta) We can find z^4 by converting 8(3^0.5+i) (cartesian form) into exponential form by finding the modulus and argument of this: (I will do the working of this question on the whiteboard when asked) z^4=16e^i(π/6 + 2Kπ) , where K is any integer. We have needed to add 2Kπ to account for the arbitrary number of rotations; any integer K can vary the argument by 2π K times however since this is a full rotation this argument will still represent the same complex number. We know that z^4=(re^(itheta))^4=r^4e^(4itheta) hence we can compare coefficients: r^4=16 implies r=2 z^4=(re^(itheta))^4=r^4e^(4itheta) 4theta = π/6+2Kπ impllies theta = π/24 + 0.5Kπ We are not given an interval for the argument so we assume the standard interval (-π,π) and find all arguments of each root within this interval by considering all the possible values of K: k=0 case theta=1/24π k=1 case theta=13/24π k=-1 case theta =-11/24π k=-2 case theta = -23/24π Hence we can conclude the four complex roots in exponential form are: 2e^(i1/24π), 2e^(i13/24π), 2e^(i-11/24π), 2e^(i*-23/24π)