For what values is the expression always real?

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Homework Statement


I have the expression ##a_2 \overline{\beta} + a_3 \beta## involving complex variables, where ##|\beta| \le 1##. I was wondering, is it possible to determine for what ##a_2## and ##a_3## the expression is always real, for every ##\beta## satisfying ##|\beta| \le 1##; or is there insufficient information?

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Thread 'Finding the nth roots of a complex number'

There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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