If $\alpha$ and $\beta$ are the roots of the equation $x^2\, -\, 2x\, +\, 4\, =\, 0$, then $\alpha ^9\, +\, \beta ^9\, =$

The roots of x^2 - 2x + 4 = 0 are 1 +/- i*sqrt(3), which can be written as 2(cos(pi/3) +/- i*sin(pi/3)). By De Moivre's theorem, alpha^9 + beta^9 = (2*e^(i*pi/3))^9 + (2*e^(-i*pi/3))^9 = 2^9 * (e^(i*3*pi) + e^(-i*3*pi)) = 2^9 * (cos(3*pi) + i*sin(3*pi) + cos(-3*pi) + i*sin(-3*pi)) = 2^9 * (-1 + 0 - 1 + 0) = -2 * 2^9 = -2^10.