x² + x + 1 = 0   , then the value  of    ( x³ + 1 / x³ )³    is ???????? 

roots of the given eqn are w,w^2 so x^3 = 1 also x^-3 = 1 therefore ans = 2^3 = 8

Since, the equation has imaginary roots.
So, on solving we get roots as omega(w) and w^2.
we know that w^3=1.
So, ans comes out to be ((w^3 +1)/(w^3))^3.
hence, ans is 8.

x^2 + x + 1 = 0

The roots of the equation are

 -(-1)^(1/3)and (-1)^(2/3)

Now for x = -(-1)^(1/3)

( x³ + 1 / x³ )  = 2

or ( x³ + 1 / x³ )³  = 2³ = 8 

Similarly for x = -(-1)^(2/3)

Then

( x³ + 1 / x³ )  = -2

or ( x³ + 1 / x³ )³  = 8