Complex numbers

If z not equal 0 is a complex number, then what is amp (z) + amp (z^-) equal to?

If then what is the value of (1 + α¹⁹ - α³⁵)100 - (1 - 3α²⁵ + α³⁸)⁵⁰?

If where i √-1, then what is the smallest positive value of (m - n)?

Let 1, ω,ω² be three cube roots of unity. If x = a + b, y = aω+ bω², z = aω² + bω, then what is x² + y² +z² equal to?