If w is a positive number and `w^(2)=2`, then `w^(2)=`

If omega is a complex number such that omega ^(3) =1, then the value of (1+ omega -omega^(2))^(4)+(1+ omega ^(2)-omega )^(4) is-

If omega is a complex cube root of unity, then (2-omega)(2-(omega)^2)(2-(omega)^10)(2-(omega)^11) is

If omega is a non-real cube root of unity, then (1+2omega+3omega^2)/(2+3omega+omega^2)+(2+3omega+omega^2)/(3+omega+2omega^2) is equal to

Let w be the complex number cos(2pi)/3 + isin(2pi)/3 . Then the number of distinct complex numbers z satisfying |(z+1, w, w^2),(2, z+w^2, 1),(w^2, 1, z+w)|=0 is equal

For complex numbers z and w, prove that |z|^(2)w-|w|^(2)z=z-w, if and only if z=w or zbar(w)=1.

If w is a complex cube root of unity, show that ([[1,w,w^2],[w,w^2,1],[w^2,1,w]]+[[w,w^2,1],[w^2,1,w],[w,w^2,1]])*[[1],[w],[w^2]]=[[0],[0],[0]]

If omega is a complex cube root of unity, show that ([[1,omega,omega^2],[omega,omega^2, 1],[omega^2, 1,omega]]+[[omega,omega^2, 1],[omega^2 ,1,omega],[omega,omega^2, 1]])[[1,omega,omega^2]]=[[0, 0 ,0]]

If omega is an imaginary cube root of unity, then the value of the determinant |(1+omega,omega^(2),-omega),(1+omega^(2),omega,-omega^(2)),(omega+omega^(2),omega,-omega^(2))| is

If omega is an imaginary cube root of unity, then the value of the determinant /_\ = |[1+omega,omega^2,-omega],[1+omega^2,omega,-omega^(2)],[omega+omega^2,omega,-omega^(2)]| is

If omega is an imaginary cube root of unity, then the value of the determinant |(1+omega,omega^2,-omega),(1+omega^2,omega,-omega^2),(omega+omega^2,omega,-omega^2)|