Five roots

Algebra Level 5

The equation x52x41=0x^5-2x^4-1=0 has five complex roots r1,r2,r3,r4,r5r_1, r_2, r_3, r_4, r_5. Find the value of 1r18+1r28+1r38+1r48+1r58.\frac{1}{r_1^8}+\frac{1}{r_2^8}+\frac{1}{r_3^8}+\frac{1}{r_4^8}+\frac{1}{r_5^8}.

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