If z^2+z+1=0 , where z is a complex number, then (z+ 1 z )^3+ (z^4+ 1 z^4 )^3 is equal to

Step-by-step solution

Given, z^2+z+1=0 aligned & (z+ 1 z )^3+ (z^4+ 1 z^4 )^3= ( z^2+1 z )^3+ ( z^8+1 z^4 )^3 & = (- z z )^3+ ( z^8+1 z^4 )^3 [ z^2+z+1=0 z^2+1=-z ] & =-1+ ( z^8+1 z^4 )^3 & z^2+1=-z , squaring both side, & z^4+1+2 z^2=z^2 z^4+1=-z^2 aligned Again, squaring it z^8+1+2 z^4=z^4 z^8+1=-z^4 Putting in Eq. (i), we get aligned & (z+ 1 z )^3+ (z^4+ 1 z^4 )^3=-1+ (- z^4 z^4 )^3 & =-1+(-1)=-2 aligned