Complex

Algebra Level 5

The 1010 complex roots of the equation z10+(13z1)10=0z^{10}+(13z-1)^{10}=0 can be partitioned into 55 pairs of complex numbers which are (a1,b1), (a2,b2), (a3,b3), (a4,b4), (a5,b5)\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 aia_i and bib_i are complex conjugates. Find the value of i=151aibi.\sum_{i=1}^5 \dfrac{1}{a_i b_i}.

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