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 \[ x^7 = 1?\]

What is the product of the fourth roots of unity?

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

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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