Newtons Inverse Roots?

Algebra Level 5

If r1,r2,r3,r4,r5r_1, r_2, r_3, r_4, r_5 are the complex roots of the equation x53x41=0x^5-3x^4-1=0 . Find the value of 1r19+1r29+1r39+1r49+1r59.\frac{1}{r_1^9}+\frac{1}{r_2^9}+\frac{1}{r_3^9}+\frac{1}{r_4^9}+\frac{1}{r_5^9}.

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