Show that the sum of the n nth roots of any complex number is zero.

To prove that the sum of the nth roots of any complex number is zero.

Complex numbers possess real numbers and imaginary numbers; a complex can be written in the form of:

$z=x+iy$

Here x and y are real numbers, and i is the imaginary number which is known as iota, whose value is $\sqrt{-}1$ .

Let's assume a complex number z then z = r$\times \exp \left({\theta }_i\right)$ ……. (1) Write the Roots and angle of the number.

$z_k={\left[rexp\left({\theta }_k\right)\right]}^{\frac{1}{n}}$ ……. (2)

${\theta }_k=\frac{\theta +2n\pi k}{n}$ ……. (3)

Substitution in (2) from (3) as:

$$\begin{gathered} z_k=r^{\frac{1}{n}}\left[exp\left(\frac{\theta i+2n\pi ki}{n}\right)\right] \\ =\left[r^{\frac{1}{n}}exp\left(\frac{\theta i}{n}\right).exp\left(\frac{2n\pi ki}{n}\right)\right] \\ =Cexp\left(\frac{2n\pi ki}{n}\right)Wherek=0,1,2,3,... \\ \mathrm{Take}\underset{}{\sum \mathrm{of}\mathrm{the}\mathrm{above}\mathrm{equation}\mathrm{as}:} \end{gathered}$$

$$\begin{gathered} \sum _{}{z}_k=C\sum _{k=0}^{n-1}exp{\left(\frac{2n\pi ki}{n}\right)}^n \\ =C+Cexp\left(\frac{2n\pi ki}{n}\right)+Cexp{\left(\frac{2n\pi ki}{n}\right)}^2+... \\ =C\left(1+exp\left(\frac{2n\pi ki}{n}\right)+exp{\left(\frac{2n\pi ki}{n}\right)}^2+...\right) \\ =C\left(1+\omega +{\omega }^2+{\omega }^3+...\right) \\ \sum _{}{z}_k=C\sum _{k=0}^{n-1}exp{\left(\frac{2n\pi ki}{n}\right)}^n \\ =C+Cexp\left(\frac{2n\pi ki}{n}\right)+Cexp{\left(\frac{2n\pi ki}{n}\right)}^2+... \\ =C\left(1+exp\left(\frac{2n\pi ki}{n}\right)+exp{\left(\frac{2n\pi ki}{n}\right)}^2+...\right) \\ 1+\omega +{\omega }^2+{\omega }^3+...\mathrm{is}\mathrm{roots}\mathrm{of}\mathrm{number}1. \end{gathered}$$

The above series is a geometrics series, put in the form of:

$\sum _{}{z}_k=C\left(\frac{{\omega }^n-1}{\omega -1}\right)$ …….(6)

$$\begin{gathered} \omega =\sqrt[n]{1} \\ \omega =exp\left(2\pi i/n\right) \\ {\omega }^n=1 \end{gathered}$$

Substitution in (6) yield$\sum _{}{z}_k=0$ .

Hence, proved that the sum of the n nth roots of any complex number is zero.