(a) Let \(y = x^{-3}\), then \(x = y^{-\frac{1}{3}}\). Substitute into the original equation:

\(2y^{-1} + 5y^{-\frac{2}{3}} - 6 = 0\)

Multiply through by \(y\) to eliminate the fractions:

\(5y^{-\frac{2}{3}} = 6 - 2y^{-1}\)

\(5y^{-\frac{2}{3}} = 6 - 2y^{-1}\)

\(125y = (6y - 2)^3\)

Expanding and simplifying gives:

\(216y^3 - 216y^2 - 53y - 8 = 0\)

(b) Using the identity \(\alpha^{-3} + \beta^{-3} + \gamma^{-3} = 1\) and \(\alpha^{-3}\beta^{-3} + \beta^{-3}\gamma^{-3} + \gamma^{-3}\alpha^{-3} = -\frac{53}{216}\), we find:

\(\alpha^{-6} + \beta^{-6} + \gamma^{-6} = (\alpha^{-3} + \beta^{-3} + \gamma^{-3})^2 - 2(\alpha^{-3}\beta^{-3} + \beta^{-3}\gamma^{-3} + \gamma^{-3}\alpha^{-3})\)

\(\alpha^{-6} + \beta^{-6} + \gamma^{-6} = 1^2 - 2\left(-\frac{53}{216}\right)\)

\(\alpha^{-6} + \beta^{-6} + \gamma^{-6} = \frac{161}{108}\)

(c) Using the equation \(216S_9 = 216S_6 + 53S_3 + 24\) and substituting the values from parts (a) and (b):

\(S_9 = \frac{399}{216} = \frac{133}{72}\)