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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.

How many non-real solutions are there to the equation \[ x^7 = 1?\]

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What is the product of the fourth roots of unity?

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How many of the 12th roots of unity are not 4th roots of unity?

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How many of the 5th roots of unity have positive real part?

**Note:** The real part of a complex number that can be written as \(a + bi\) for real numbers \(a\) and \(b\) is \(a\).

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