Question

how to find cube root of unity?​

Answer 1

Let the cube root of 1 be x i.e., 3√ 1 = x.


Then by definition, x3 = 1 or x3 � 1 = 0 or (x � 1) (x2 + x + 1) = 0


Either x � 1 = 0 i.e., x = 1 or (x2 + x + 1) = 0


Hence 


Hence, there are three cube roots of unity which are  which the first one is real and the other two are conjugate complex numbers. These complex cube roots of unity are also called imaginary cube roots of unity.


PROPERTIES OF THE CUBE ROOTS OF UNITY:

Answer 2

We will discuss here about the cube roots of unity and their properties.

Suppose let us assume that the cube root of 1 is z i.e., ∛1 = z.

Then, cubing both sides we get, z3 = 1

or, z3 - 1 = 0

or, (z - 1)(z2 + z + 1) = 0

Therefore, either z - 1 = 0 i.e., z = 1 or, z2 + z + 1 = 0

Therefore, z = −1±12−4⋅1⋅1√2⋅1 = −1±−3√2 = -12 ± i√32

Therefore, the three cube roots of unity are

1, -12 + i√32 and -12 - i√32

among them 1 is real number and the other two are conjugate complex numbers and they are also known as imaginary cube roots of unity.