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Algebra Level 4

Let \(\alpha\) and \(\beta\) be the roots of \(x^2 - 6x - 2 = 0\), with \(\alpha > \beta \). If \(a_n = \alpha ^n - \beta ^n \) for \(n\ge 1\), then find the value of:

\[\frac {a_{10} - 2a_8} {2a_9} \]

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