Question

Write two complex cube root of 1 .

Answer 1

W,W^2 are two complex cube roots of unity

(pronounced as omega,omega square )

W = [ -1+sqrt (-3)]/2

W^2 = [ -1 - sqrt (-3)]/2

sqrt (-1) = I

Hope it helps . . . . . . . . .

Answer 2

The three cube roots of unity are:

e three cube roots of unity are:1

e three cube roots of unity are:1-1/2+i√(3)/2 , and

e three cube roots of unity are:1-1/2+i√(3)/2 , and -1/2 – i√(3)/2

Given :

The number 1

To find :

Two complex cube roots of 1

Solution :

Let the cube root of 1 be =a

That is ,

$\sqrt[3]{1} = a$

According to the general cube roots definition,

a^3 = 1 or a^3 – 1 = 0

(a^3 – b^3) = (a – b) ( a^2 + ab + b^2)

Now, (a^3 – 1^3) = 0

=> (a – 1)( a^2 + a + 1) = 0

Therefore, a = 1

or

( a^2 + a + 1) = 0

By using quadratic roots formula for the above equation, we get;

a = [(-1) ± √(1^2-4.1.1)]/2

= [-1 ± √-3]/2

= -1/2 ± i√(3)/2

Therefore, the three cube roots of unity are:

1

-1/2+i√(3)/2 , and

-1/2 – i√(3)/2

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