Assume z1 = a1+b1i = r1,θ1, and z2 = a2+b2i = r2,θ2.
The new radius is r1r2, and the new angle is θ1+θ2. Convert the product back to rectangular coordinates as follows.
x ← r1r2×cos(θ1+θ2)
y ← r1r2×sin(θ1+θ2)
Expand using the angle addition formulas. Replace r?cos(θ?) and r?sin(θ?) with a1 a2 b1 b2 appropriately and get x = a1a2-b1b2 and y = a1b2+a2b1. This is the formula for multiplication of complex numbers in rectangular coordinates, as presented in the introductory page. Therefore Demoivre's formula is valid.
This formula provides efficient procedures for complex exponentiation. If z is 3 units from the origin, at 45°, then z4 is -81. Reverse this process to take roots. The 6 sixth roots of 64 all have radius 2, lying on a circle 2 units from the origin. Every angle that is a multiple of 60°, when multiplied by 6, becomes a multiple of 360°, and lands on the positive x axis, which is what we want. Thus the 6 roots are at angles 0° 60° 120° 180° 240° 300°, radius 2. You are familiar with 2 and -2, you may not know the others as well. The root at 60° is 1+sqrt(3)i.
A sixth degree polynomial has at most 6 roots, so we've found them all. In other words, the polynomial x6-64 splits, using the roots identified above.