SOLUTION: Show that z^n +(complex conjugate of z)^n is always real for integer n.

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Question 1160752: Show that z^n +(complex conjugate of z)^n is always real for integer n.
Answer by math_helper(2461) About Me  (Show Source):
You can put this solution on YOUR website!



x = a+bi  = Me%5E%28i%2Atheta%29 

x_ = a-bi = Me%5E%28-i%2Atheta%29



(x_ is complex conjugate of x)



where M=sqrt%28a%5E2%2Bb%5E2%29 and theta+=+tan%5E-1%28b%2Fa%29+



x%5En+ = 

x_%5En+ = 

=  +M%5En%2Acos%28theta%2An%29+-+i%2AM%5En%2Asin%28theta%2An%29+



 =  +blue%282%2AM%5En%2Acos%28theta%2An%29%29++



Notice +blue%282%2AM%5En%2Acos%28theta%2An%29%29++  has no imaginary component, hence it is strictly real.