Oh!!!Come On!!!!

The roots of the equation will be -w(omega) & $-w^{2}$ ..

Let alpha = -w & beta = $-w^{2}$ ...

so $(alpha)^{2009}$ will be $-w^{2009}$ = - $w^{2007}$ $*$ $w^{2}$

since 2007 is a multiple of 3 and $w^{(multiple of 3)}$ is 1 so $w^{2007}$ =1.

..so $(alpha)^{2009}$ =- $w^{2 }$ .

..Now $(beta)^{2009}$ = - $w^{4018}$ = - $w^{4017}$ w

again 4017 is a multiple of 3 so $w^{4017}$ =1..

..so $(beta)^{2009}$ = -w.

..therefore $alpha^{2009}$ + $beta^{2009}$ = - $w^{2}$ - w =(- $w^{2}$ +w) = -(-1) ( since $w^{2}$ +w+1=0) =1.

..so $\frac{pie}{a}$ =1 so a= pie =3.14...

The roots of the equation will be : -w(omega) & -w^2(omega square).. Let alpha = -w & beta = -w^2....so (alpha)^2009 will be -w^2009 = - (w^(2007) *w^2 ) since 2007 is a multiple of 3 and w^(multiple of 3) is 1 so w^2007 =1...so (alpha)^2009 =-w^2 ...Now (beta)^2009 = -w^4018 = - w^4017 * w again 4017 is a multiple of 3 so w^4017 =1....so (beta)^2009 = -w...therefore alpha^(2009) + beta^(2009) = -w^2 - w = -(w^2 +w) = -(-1) ( since w^2+w+1=0) =1...so pie/a =1 so a= pie =3.14...