For any complex number `w=a+bi,` where `a, b in R.` If `w = cos 40^@ + i sin40^@,` then| `+2w^2+3w^3+......+9w^9|^-1` equals

For any complex number w=a+bi, where a,b in R. If w=cos40^(@)+i sin40^(@), then |+2w^(2)+3w^(3)+......+9w^(9)|^(-1) equals

If w is a complex cube root of unity, then show that (1-w) (1- w ^(2)) (1 - w ^(4)) ( 1 - w ^(5)) = 9

If omega ne 1 is a complex cube root of unity, then 5.23 + omega + omega^(((1)/(2) + (3)/(8) + (9)/(32) + (27)/(128)+...)) is equal to __________

Let omega be a complex number such that 2omega+1=z where z=sqrt(-3) . If [(1,1,1), (1, -omega^(2)-1 , omega^(2)),(1, omega^(2) , omega^(7))]=3k , then k is equal to

If omega is a complex cube root of unity, then (1-omega+(omega)^2)^3 =

If i_(1) = 3 sin omega t and i_(2) =6 cos omega t , then i_(3) is

If omega is a complex cube root of unity, then find the values of: (1 - omega - omega^2)^3 + (1 - omega + omega^2)^3

Let omega ne 1 be cube root of unity. Suppose (1 + omega)^(2023) = A + B omega where A, B in R, then |A + Bi|^(2) is equal to ______________.