# nth roots of a complex number

## nth Roots of a complex number

Let z be a complex number which has the following polar form,

`z = r(cos\theta + i * sin\theta)`,

r = |z| is the modulus of z.

`\theta` is the argument of z.

To calculate the polar form, use this calculator:

calculate the polar form of a complex number

z has exactly n nth roots here denoted by `t_k` with `0 <=k<=n-1`,

`t_k = \text{}^nsqrt(r)(cos((\theta+2 \pi k)/n) + i * sin((\theta+2 \pi k)/n))`

To proove that, we use de Moivre's formula which states, for every integer n,`(cos\alpha+i*sin\alpha)^n = cos(n*\alpha) + i*sin(n*\alpha)`

We apply this formula to `t_k` :

`t_k^n = (\text{}^nsqrt(r))^n(cos(n*(\theta+2 \pi k)/n) + i * sin(n*(\theta+2 \pi k)/n))`

`t_k^n = r(cos(theta) + i * sin(\theta)) = z`

`t_k` is a n-th root of z.

## See also

Polar form of a complex number

Modulus of a complex number

Argument of a complex number