A root of unity is a complex number that, when raised to a positive integer power, results in 1. Roots of unity have applications to the geometry of regular polygons, group theory, and number theory.

If \(w \not= 1\) is an \(n\)-th root of unity, then find the value of

\[ 1+w+w^2+w^3+\cdots +w^{n-1}\]

Evaluate

\[ \cos \frac{ \pi}{7} - \cos \frac{ 2\pi }{7} + \cos \frac{ 3 \pi}{7} . \]

\[\huge x^{1729}+x^{-1729}\]

Find the value of the above expression if \(x + \frac 1 x = 1 \)

Given \(z^2+z+1=0,\) find the value of

\[\left(z+\dfrac{1}{z} \right)^2+\left(z^2+\dfrac{1}{z^2}\right)^2+\left(z^3+\dfrac{1}{z^3} \right)^2+\cdots+\left(z^{21}+\dfrac{1}{z^{21}}\right)^2.\]

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