# Three Cube Roots Of Unity

The cube **roots**** **of unity are the cube roots of one.

**Example:**

**Proof: **

Let x be the cube roots of unity

X^{3}=1

X^{3}-1=0

X^{3}-1^{3}=0 1^{3}=1

By using

a^{3 }– b^{3}= (a-b) (a^{2}+ab+b^{2})

X^{3} -1^{3}=(x-1) (x^{2}+x+1^{2})

(x – 1) (x^{2} + x+1^{2})=0

X – 1=0 x^{2}+x+1=0

X=1 x^{2}+x+1=0

By using quadratic formula

Thus the three cube roots of unity are:

We know that I am called a complex number.

## Properties of cube roots of unity

- Each complex cube root of unity is square of the other
- The Sum of cube roots of unity is zero
- Product of cube roots of unity is one
- Each complex cube root of unity is reciprocal to other
- Each complex cube root of unity is conjugate of each other
- Each complex cube root of unity is the multiplicative inverses of each other
- For an is equivalent to one of the cube roots of unity

**Each complex cube root of unity are square of the other**

Complex cube roots of unit

**Product of cube roots of unity is one**

**Proof**:

**Each complex cube root of unity is reciprocal of each other**

**Proof:**

As we know

Each complex cube root of unity is reciprocal to each other.

**Each complex cube root of unity is conjugate of each other**

**Proof**:

Let

Hence proved

Each complex cube root of unity is conjugate of each other

**For any ****equivalent to one of the cube roots of unity.**

## Frequently Asked Question-FAQs

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

The cube roots of unity are the numbers which give a result of 1 when raised to the power of 3. In other words, the cube root of unity is the cube root of 1, or 3âˆš1.

### What are the Values of Cube Roots of Unity?

The cube root of unity values are 1, âˆ’Â½ + i âˆš(3/ 2), and âˆ’Â½ â€“ i âˆš(3/ 2)

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

According to the properties of the cube root of 1, the sum of its root is zero. So, 1 + Ï‰ + Ï‰^{2}=0.

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

According to the properties Product of cube roots of unity is one

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