If $$z$$ is a nonreal root of $$\root 7 \of { - 1} $$  then $${z^{86}} + {z^{175}} + {z^{289}}$$    is equal to

Solution :
$$\eqalign{ & {z^7} = - 1 \cr & \therefore \,\,{\text{expression}} = {\left( {{z^7}} \right)^{12}} \cdot {z^2} + {\left( {{z^7}} \right)^{25}} + {\left( {{z^7}} \right)^{41}} \cdot {z^2} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\left( { - 1} \right)^{12}} \cdot {z^2} + {\left( { - 1} \right)^{25}} + {\left( { - 1} \right)^{41}} \cdot {z^2} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {z^2} - 1 - {z^2} = - 1. \cr} $$