Question 4138
 Z=1+i√3, note r = √(a^2+b^2) = √(1+3) = 2, 
 and theta = Arc Tan(b/a) = Arc Tan(√3) = pi/3 (set theta = x)
 in polar coordinates, z = r(cos x + i sin x)
                         = 2(cos pi/3 + i sin pi/3)
 By DeMoivre Theorem,
 z^n = 2^n(cos pi/3 + i sin pi/3)^n = 2^3(cos n pi/3 + i sin n pi/3).
 If z^n is real, then sin npi/3 = 0 , equivalently n pi /3 must
 be multiple of pi. We see that when n = 3 , n pi /3 = pi.
 Hence,the smallest positive integer n such that z^n is real is 3 and
 we have z^3 = 2^3(cos pi + i sin pi) = 8(-1+ i * 0) = -8. 
 
 Note that if z^n is imaginary then cos n pi/3 should be 0 and so 
 n pi/3 must be equal to (2k +1)pi/2 for some integer k.
 But n pi/3 = (2k +1)pi/2 implies 2n = 3(2k+1), which is impossible
 for any integer because the left side is even while the right hand side is
 odd. Hence,z^n cannot be iaginary.
 This completes the proof of the two requirements.

 Kenny