Roots of complex numbers in polar form. The n distinct n-th roots of the complex number

           z = r( cos θ + i sin θ)

can be found by substituting successively k = 0, 1, 2, ... , (n-1) in the formula

Derivation. Let z = r( cos θ + i sin θ) and w = ρ(cos φ + i sin φ) be an n-th root of z. Then by De Moivre’s formula

1]        ρⁿ(cos nφ + i sin nφ) = r( cos θ + i sin θ) .

Consequently,

3]        ρⁿ = r

4]        cos nφ = cos θ

5]        sin nφ = sin θ .

Thus

What values of φ will satisfy the conditions of 4] and 5] above? Well, 4] and 5] are satisfied if and only if

7]        nφ = θ + k·360^o

or, equivalently,

8]        φ = (θ + k·360^o ) / n

where k = 0, 1, 2, ...

Therefore, any n-th root of z must be one of the numbers

where k = 0, 1, 2, ...