Question 1160752
<pre>

x = a+bi  = {{{Me^(i*theta)}}} 
x_ = a-bi = {{{Me^(-i*theta)}}}

(x_ is complex conjugate of x)

where M={{{sqrt(a^2+b^2)}}} and {{{theta = tan^-1(b/a) }}}

{{{x^n }}} = {{{ M^n*e^(i*theta*n) = M^n*cos(theta*n) + i*M^n*sin(theta*n) }}}
{{{x_^n }}} = {{{ M^n*e^(-i*theta*n) = M^n*cos(theta*n) + i*M^n*sin(-theta*n) }}}
=  {{{ M^n*cos(theta*n) - i*M^n*sin(theta*n) }}}

{{{x^n+x_^n =  M^n*cos(theta*n) + i*M^n*sin(theta*n) + M^n*cos(theta*n) - i*M^n*sin(theta*n) }}} =  {{{ blue(2*M^n*cos(theta*n))  }}}

Notice {{{ blue(2*M^n*cos(theta*n))  }}}  has no imaginary component, hence it is strictly real.