Multiplying the polar form of the complex number:

**z =|z| (cos Ө + i sin Ө)**

by itself n times then we multiply the size **|z|** by itself n times, giving the final size **|z| ^{n}**. We add the angle Ө to Ө, n times, giving the final angle nӨ. The resultant polar form is called de Moivre’s formula (also known as de Moivre’s theorem),

z^{n }=**|z| ^{n }**

**(cos Ө + i sin Ө)**

where n may be positive, negative, integer, or fractional, although if n is fractional then the result is multi-valued. I will give the two most common examples of the use of de Moivre’s theorem, the square root and the cube roots of of a complex number,

There are two square roots of a complex number separated by 180°, i.e. 360°/2, as in Fig. 3.10. All you need to do is find the first root (the principal root) and then add another root at 180° to the first square root.

The second Example of the n being fractional is the cube root of :

**z = 27 (****cos 120 ^{0}**

**+ i sin 120**

^{0)}As z is identical for 120° and 120° plus either one or two revolutions i.e. (120° + 360° = 480°) and (120° + 360° + 360° = 840°) we have,

There are three cube roots of a complex number each separated by 120°, i.e. 360°/3, as in below Figure. All you need to do is find the first root and then add 120° twice for the other two cube roots.