Let α and α2 be the roots ofx2+x+1=0 then the equation whose roots are α31 and α62, is

Since, α,α2 be the roots of the equation x2+x+1=0∴     α+α2=−1 and     α3=1    ...(i)  Now,     α31+α62=α311+α31⇒    α31+α62=α30⋅α1+α30⋅α⇒    α31+α62=α310⋅α1+α310⋅α⇒    α31+α62=α(1+α)     [from Eq. (ii)] ⇒    α31+α62=−1    [ from Eq. (i)]  Again,      α31⋅α62=α93⇒    α31⋅α62=α331=1∴ Required equation is,x2−α31+α62x+α31⋅α62=0⇒x2+x+1=0