Dear student,

The cube roots of unity are 1, w, w^2.
And according to the property of cube roots of unity :- 1 + w + w^2 = 0

Now, if alpha and beta are imaginary cube roots of unity, so take alpha as w and beta as w^2 ( since 1 is a real root ).

Now, (alpha)^4 + (beta)^4 +(alpha beta) = w^4 + (w^2)^4 + (ww^2).

Now, we get :-
w^4 + w^6 + w^3

Taking w^3 as common we get :-

w^3 ( 1 + w + w^2 )

And 1 + w + w^2 = 0.

So, the value of above equation is 0.
Therefore answer is = 0.