Complexity Is Awesome!

Algebra Level 5

The equation x10+(13x1)10=0x^{10}+(13x-1)^{10}=0 has 1010 complex roots: r1,r1,r2,r2,r3,r3,r4,r4,r5,r5,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.

1r1r1+1r2r2+1r3r3+1r4r4+1r5r5\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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