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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