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# Three Cube Roots Of Unity

June 25, 2022
written by Azhar Ejaz

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

Example:

Proof:

Let x be the cube roots of unity

X3=1

X3-1=0

X3-13=0                        13=1

By using

a3 – b3= (a-b) (a2+ab+b2)

X3 -13=(x-1) (x2+x+12)

(x – 1) (x2 + x+12)=0

X – 1=0                 x2+x+1=0

X=1                            x2+x+1=0

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.

### 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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