Roots of a complex number

Let \(n\) be a positive integer and let \(w=r(\cos\theta+i\sin\theta)\), with \(r>0\). The solutions of \(z^n=w\) are \[z_k=r^{1/n}\left[\cos\left(\frac{\theta+2k\pi}{n}\right)+i\sin\left(\frac{\theta+2k\pi}{n}\right)\right],\quad k=0,1,\ldots,n-1.\] If instead \(w=0\), the only distinct root is \(z=0\); its multiplicity is \(n\).

For \(r>0\), note the following:

1. There are \(n\) distinct \(n^{\mathrm{th}}\) roots of \(r(\cos\theta + i\sin\theta)\)
2. Each root has modulus \(r^{\frac{1}{n}}\).
3. One of the roots has argument \(\frac{\theta}{n}\).
4. Successive arguments differ by \(\frac{2\pi}{n}\).
5. For \(n\ge2\), the sum of the \(n\) roots is \(0\). For \(n=1\), the sole root is \(w\).
6. On an Argand diagram, the points representing the \(n^{\mathrm{th}}\) roots of \(r(\cos\theta + i\sin\theta)\) lie on a circle of radius \(r^{\frac{1}{n}}\) and are separated by angles \(\frac{2\pi}{n}\).
7. Unless \(\theta\) is a multiple of \(\pi\), the roots do not occur in conjugate pairs.