Cube Roots Of Unity

The non-real cube roots of unity satisfy 1 + omega + omega^2 = 0 and omega^3 = 1, identities that drive most JEE Advanced complex-number simplifications. From the first identity, 1 + omega = -omega^2 and 1 + omega^2 = -omega. Therefore (1 + omega)^3 = (-omega^2)^3 = -omega^6 = -(omega^3)^2 = -1, and (1 + omega^2)^3 = (-omega)^3 = -omega^3 = -1. The required difference is (-1) - (-1) = 0. The symmetry between omega and omega^2 makes the two cubes equal, forcing the difference to vanish. Option 2 ignores that both cubes equal -1. Option -2 reverses a sign incorrectly. Option 1 drops a factor. Hence the expression equals 0. Plausibility check: since omega and omega^2 are complex conjugates, the two bracketed terms are conjugates, and cubing preserves conjugacy; their difference is twice the imaginary part of a real number, which is zero, confirming the result. The cube roots of unity sit at the vertices of an equilateral triangle inscribed in the unit circle, so their geometric symmetry mirrors the algebraic relation that their sum vanishes. Exploiting omega and omega squared as a conjugate pair, and repeatedly reducing powers using omega cubed equal to one, collapses a wide variety of intimidating expressions to small real numbers with very little computation.