Question

What are the values of the cube roots of unity?

Answer 1

Determine the values of the cube roots of unity.

A root of unity is a complex number that returns $1$ when raised to the power of a positive integer $n$. These roots are applied in several fields and themes of mathematics, such as number theory.

In the following, use De Moivre's Theorem to calculate the cube roots of unity in detail:

The cube roots of unity are also the integers that, one is raised to the power of three, provide the value$1$.
As said, the cube root of unity is the cube root of $1$ that is$\sqrt[3]{1}$. All values of the cube root of unity are$1,-\frac{1}{2}+i\sqrt{\left(\frac{3}{2}\right)},\text{and}-\frac{1}{2}–i\sqrt{\left(\frac{3}{2}\right)}$.

Hence, cube root of unity are$1,-\frac{1}{2}+i\sqrt{\left(\frac{3}{2}\right)},\text{and}-\frac{1}{2}–i\sqrt{\left(\frac{3}{2}\right)}$.