Find value of $(\cos\frac{2\pi}{7})^ {\frac{1}{3}} + (\cos\frac{4\pi}{7})^ {\frac{1}{3}} + (\cos\frac{8\pi}{7})^ {\frac{1}{3}} $ This question was on my list. I was trying to apply the $n$-th roots of unity, but other ideas are welcome. I also tried Newton's sums, but it's not working.
I searched around here and I didn't find a similar one, but if they do, just say that I delete the topic.

If you begin with double your cosines, you have the three (real) roots of $$ x^3 + x^2 - 2x - 1 $$ so that the cosines themselves are the three roots of
$$ 8 w^3 + 4 w^2 - 4 w - 1  $$
After that there is still a bunch of fiddling, since your numbers are the cube roots of these. 
From 1875 book by Reuschle. The original technique is due to Gauss, before Galois.  A modern treatment is in Galois Theory by Cox.

