Question 13
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. Find the value of $$
\begin{aligned}&\frac1{r_1\overline{r_1}}+\frac1{r_2\overline{r_2}}+\frac1{r_3\overline{r_3}}\\&\quad+\frac1{r_4\overline{r_4}}+\frac1{r_5\overline{r_5}}.\end{aligned}
$$
x^{10}+(13x-1)^{10}=0
$$ has $10$ complex roots $r_1,$ $\overline{r_1},$ $r_2,$ $\overline{r_2},$ $r_3,$ $\overline{r_3},$ $r_4,$ $\overline{r_4},$ $r_5,$ $\overline{r_5},$ where the bar denotes complex conjugation. Find the value of $$
\begin{aligned}&\frac1{r_1\overline{r_1}}+\frac1{r_2\overline{r_2}}+\frac1{r_3\overline{r_3}}\\&\quad+\frac1{r_4\overline{r_4}}+\frac1{r_5\overline{r_5}}.\end{aligned}
$$