In this explainer, we will learn how to identify the cubic roots of unity using de Moivre’s theorem.

A cube (or cubic) root of unity is a complex-valued solution to the equation =1. If we only consider real-valued solutions to this equation, we can apply the cube root to both sides of the equation to obtain =√1=1, which means that there is only one real-valued solution. However, there are more solutions to this equation that are not real numbers. To solve this equation algebraically, we first rearrange this equation as −1=0. Recalling the difference of cubes formula − =( − ) + + , we can factor −1 to write ( −1) + +1=0.

The first factor −1 leads to the real root =1, while the quadratic factor + +1 will lead to complex-valued roots when we apply the quadratic roots formula. Applying the quadratic roots formula with =1, =1, and =1, we have =− ±√ −4 2 =−1±√1−42=−12±√32 .

This gives us two other complex-valued solutions to the equation =1. Hence, we have obtained three cube roots of unity.

### What is the Definition of Cube Root of Unity?

The cube roots of unity can be defined as the numbers which when raised to the power of 3 gives the result as 1. In simple words, the cube root of unity is the cube root of 1 i.e.3√1.

## What are Cube Roots of Unity?

There are a total of three cube roots of unity which are as follows:

Cube Root of Unity Value | Nature of Cube Root |
---|---|

1 | Real |

−½ + i √(3/ 2) | Complex |

−½ – i √(3/ 2) | Complex |

Here, a = 1 is the real cube root of unity while a = – ½ + i √(3/ 2) and a = – ½ – i √(3/ 2) are the imaginary or complex cube roots of unity.

### Property: Square of Complex Cubic Roots of Unity

Let be a complex cubic root of unity. Then, = .

Let us consider an example where we can apply this property to simplify an expression involving .

## FAQ

**How do you find the roots of unity?**

It is defined as **the number that can be raised to the power of 3 and result is 1**. The sum of the three cube roots of unity is zero i.e., 1++2=0.

**Who discovered cube root of unity?**

**Hero of Alexandria**devised a method for calculating cube roots in the 1st century CE. His formula is again mentioned by Eutokios in a commentary on Archimedes.