The nth roots of unity are: $w_k$ = e^$\frac{2*pi*i*k}{n}$ for k = 0, 1, 2, ..., n-1.

Properties:

1. They form vertices of a regular n-gon inscribed in the unit circle
2. Sum of all nth roots = 0: $w_0$ + $w_1$ + ... + $w_{n-1}$ = 0
3. Product of all nth roots = (-1)^(n+1)
4. Sum of squares: $w_0^2$ + $w_1^2$ + ... + $w_{n-1}^2$ = 0 (for n >= 3)
5. $x^n$ - 1 = (x - $w_0$)(x - $w_1$)...(x - $w_{n-1}$)
6. 1 + x + $x^2$ + ... + $x^{n-1}$ = $\frac{x^n - 1}{(x - 1)}$ = product of (x - $w_k$) for k = 1, ..., n-1

For JEE: The most commonly tested cases are n = 3 (cube roots) and n = 4 (fourth roots = {1, i, -1, -i}). For n = 6: sixth roots are {1, w, $w^2$, -1, -w, -$w^2$} where w = e^$\frac{i*pi}{3}$.

Application: To evaluate sums like sum of cos$\frac{2*pi*k}{n}$: it equals the real part of the sum of nth roots, which is Re(0) = 0.