16. \( \frac{a+b \omega+c \omega^{2}}{c+a \omega+b \omega^{2}}= \)

To solve \( \frac{a+b \omega+c \omega^{2}}{c+a \omega+b \omega^{2}} \), where \( \omega \) is a complex root of unity (specifically, \( \omega = e^{2\pi i / 3} \) and \( \omega^3 = 1 \), \( 1 + \omega + \omega^2 = 0 \)), we can interpret the expression in terms of the properties of \( \omega \). Let’s write it in a more manageable form. We know that \( \omega^2 = -1 - \omega \) and can substitute that in. For example, replace \( \omega^2 \) in both the numerator and denominator to get expressions purely in terms of \( \omega \) and constants. Then, simplify it by grouping similar terms and factor using the identities related to \( \omega \). Owing to the symmetry and cyclical nature of the roots of unity, this expression will simplify quite nicely, and you may end up finding ratios of the sums depending on the coefficients \( a, b, \) and \( c \). The final result would generally have a clean form since the nature of \( \omega \) often leads to very elegant simplifications.