(i) If `alpha`,`beta` be the imaginary cube root of unity, then show that `alpha^4+beta^4+alpha^-1beta^-1=0`

If alpha and beta are the complex cube roots of unity, show that alpha^4+beta^4 + alpha^-1 beta^-1 = 0.

if alpha and beta are imaginary cube root of unity then prove (alpha)^4 + (beta)^4 + (alpha)^-1 . (beta)^-1 = 0

If alpha and beta are the complex cube roots of unity, then prove that (1 + alpha) (1 + beta) (1 + alpha)^(2) (1+ beta)^(2)=1

If alpha , beta , gamma are the roots of x^3 -3x^2 +3x+7=0 then show that (alpha-1)/( beta-1) +(beta -1)/(gamma -1) +(gamma -1)/( alpha-1) = 3 omega where omega is complex cube of unity .

If alpha and beta are the roots of a quadratic equation such that alpha+beta=2, alpha^4+beta^4=272 , then the quadratic equation is

If alpha , beta are the roots of x^2 +x+1=0 then alpha beta + beta alpha =

If x=a + b, y= a alpha + b beta, z= a beta + b alpha , where alpha and beta are complex cube roots of unity, then show that xyz = a^(3) + b^(3)

If alpha , beta , gamma in {1,omega,omega^(2)} (where omega and omega^(2) are imaginery cube roots of unity), then number of triplets (alpha,beta,gamma) such that |(a alpha+b beta+c gamma)/(a beta+b gamma+c alpha)|=1 is

If alpha, beta be the roots of px^(2)-qx +q=0 , then show that sqrt((alpha)/(beta)) +sqrt((beta)/(alpha)) - sqrt((g)/(p))=0 .

If alpha, beta are the roots of x^(2) + x + 3=0 then 5alpha+alpha^(4)+ alpha^(3)+ 3alpha^(2)+ 5beta+3=