If z^2+z+1=0 then (z^3+ 1 z^3 )^2+ (z^4+ 1 z^4 )^2= where z = w = complex cube root of unity

Step-by-step solution

Z is a complex cube root of unity z^3=1...(i) and 1+z+z^2=0 z^2+1=-z ...(ii) Consider, aligned & (z^3+ 1 z^3 )^2+ (z^4+ 1 z^4 )^2 & = (1+ 1 1 )^2+ (z^3 z+ 1 z^3 z )^2 & =4+ [z+ 1 z ]^2 aligned ...[From (i)] aligned & =4+ [ z^2+1 z ]^2 & =4+ [ -z z ]^2 ...[From(ii)] & =4+(-1)^2 =5 aligned