Complex

Algebra Level 5

The $10$ complex roots of the equation $z^{10}+(13z-1)^{10}=0$ can be partitioned into $5$ pairs of complex numbers which are $\left(a_1,b_1\right),~\left(a_2,b_2\right),~\left(a_3,b_3\right),~\left(a_4,b_4\right),~\left(a_5,b_5\right)$ where in each pair $a_i$ and $b_i$ are complex conjugates. Find the value of $\sum_{i=1}^5 \dfrac{1}{a_i b_i}.$

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