If \frac{1}{a} + \frac{1}{b} = \frac{5}{6} and \frac{1}{a^2} + \frac{1}{b^2} = \frac{13}{36}, then what is \frac{1}{a^3} + \frac{1}{b^3} equal to?

Explanation

Squaring the first equation gives \frac{1}{a^2} + \frac{1}{b^2} + \frac{2}{ab} = \frac{25}{36}. Substituting the second gives \frac{13}{36} + \frac{2}{ab} = \frac{25}{36} \Rightarrow \frac{2}{ab} = \frac{12}{36} \Rightarrow \frac{1}{ab} = \frac{6}{36}. Then \frac{1}{a^3} + \frac{1}{b^3} = (\frac{1}{a} + \frac{1}{b})(\frac{1}{a^2} + \frac{1}{b^2} - \frac{1}{ab}) = \frac{5}{6}(\frac{13}{36} - \frac{6}{36}) = \frac{5}{6} \times \frac{7}{36} = \frac{35}{216}.

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