If $$\alpha$$ and $$\beta$$ are two roots of the equation $$x^2 + x + 1 = 0$$, then the value of $$\alpha^{2017} + \beta^{2017}$$ is

Consider, $$x^3-1$$ = (x-1)($$x^2 + x + 1 $$)=0

It will have roots as 1, $$\alpha\ $$ and $$\beta\ $$

Now, since both  $$\alpha\ $$ and $$\beta\ $$ satisfy $$x^3-1=0$$, Hence,$$\alpha^{3}-1=0$$ => $$\alpha^3=1$$

and $$\beta^3-1=0$$  => $$\beta^3=1$$

Hence $$\alpha^{2017}+\beta^{2017}$$ = $$\alpha^{3\times\ 672+1}+\beta^{3\times\ 672+1}$$  =  $$\alpha\ +\beta$$

Now, the sum of roots of the equation $$x^3-1$$=0 is zero. 

Hence,  $$1+\alpha\ +\beta\ =0=>\alpha+\beta\ =-1$$