Example 5

Using the polar form, find the quotient (1 + i)/(1 - i). In polar form,

Using the product formula (2) n times, we get a formula for the n^th power of a complex number,

(4)

(r cis θ)ⁿ = rⁿ cis (nθ).

This formula in the case r = 1 is called De Moivre's Formula, (cos θ + i sin θ)" = cos (nθ) + i sin (nθ).

We can see from the power formula (4) that the complex number r cis θ has the square root √r cis (θ/2). In fact, each complex number except zero has two square roots,

(5)