The nth roots of unity are: w_k = e^(2pii*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} = (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^(i*pi/3).

Application: To evaluate sums like sum of cos(2pik/n): it equals the real part of the sum of nth roots, which is Re(0) = 0.