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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.

# Solving

What is the complex solution set of the equation $$z^4-1=0$$?

Which of these is a solution to the equation $$z^6=12$$?

Let $$U_n$$ denote the set of all $$n^\text{th}$$ roots of unity.

If $$U_a=\{yz\mid y\in U_3,\ z\in U_5\}$$, then $$a=\ ?$$

Which of these is a solution to the following equation?

$$x^5+x^4+x^3+x^2+1=0$$

If $$a+bi$$ is an $$n^\text{th}$$ root of unity, then which of these is not necessarily an $$n^\text{th}$$ root of unity?

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