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## Roots of Unity

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.

# Level 3

$\huge x^{1729}+x^{-1729}$

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

If for $$z \in \mathbb{C} \space ,$$ $$z+\dfrac{1}{z}=2 \cos 6°.$$ Then find the value of $\left( z^{1000}+\frac{1}{z^{1000}} \right).$

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

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

The point $$\left(4+7\sqrt{3}\ ,\ 7-4\sqrt{3}\right)$$ is rotated $$\dfrac{\pi}{3}$$ radians counterclockwise about the origin. If the resulting image is $$(a,b)$$, then what is $$a+b$$?

Evaluate

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

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