In this problem, it is assumed (hidden assumption) that z is a complex number z = a + bi.
Another hidden assumption is that z* is the complex conjugate to z.
In the binomial expansion of z^n = , the imaginary parts (addends) are those and only those
that contain the factors (ib) in ODD degrees.
Similar is for the binomial expansion of (z*)^n = : the imaginary parts (addends) are those and only those
that contain the factors (-ib) in ODD degrees.
But the corresponding imaginary addends in binomial expansions z^n and (z*)^n go with opposite signs, and, THEREFORE,
they cancel each other in the sum z^n + (z*)^n.
Therefore, the sum z^n + (z*)^n is a real number.
