Evaluate $(2-w)(2-w^2)\cdots(2-w^{10})$ where $w=e^{2\pi i/11}.$
I notice that $w$ is a primitive 11th root of unity, which means that it satisfies $w^{11}=1$ and $w^k\neq 1$ for any $k$ between 1 and 10. This also means that the 11th roots of unity are $w, w^2, w^3, \ldots, w^{11}$ and they are equally spaced on the unit circle in the complex plane. I can use this to rewrite the product as $(2-w)(2-w^2)\cdots(2-w^{11})$ because $w^{11}=1$. In fact, I can get the product by plugging in $x=2$ into this factorization.
[*  ] This means that the product is equal to $2^{11}-1.$ 