If x^2+x+1=0 then prove that (x+1/x)^2+(x^2+1/x^2)^2.........+(x^27+1/x^27)^2=54

the result of the equation x^2+x+1=0 are the cuberoots of unity use that in the expression
you can write x+1/x = -1.Use this and you will get the answer