Complexity Is Awesome!

Algebra Level 5

The equation \(x^{10}+(13x-1)^{10}=0\) has \(10\) complex roots: \(r_1, \overline{r_1}, r_2, \overline{r_2}, r_3, \overline{r_3}, r_4, \overline{r_4}, r_5, \overline{r_5},\) where the bar denotes complex conjugation. Evaluate the expression below.

\[\dfrac 1{r_1\overline{r_1}}+\dfrac 1{r_2\overline{r_2}}+\dfrac 1{r_3\overline{r_3}}+\dfrac 1{r_4\overline{r_4}}+\dfrac 1{r_5\overline{r_5}} \]

Image Credit: Wikimedia Decagon by László Németh
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Parth Lohomi by Parth Lohomi
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