Rationalise the denominator 1/ √7 + √6 - √13

\frac{1}{\sqrt{7} + \sqrt{6} - \sqrt{13}} = \frac{1}{(\sqrt{7} + \sqrt{6}) - \sqrt{13}} \times \frac{(\sqrt{7} + \sqrt{6}) + \sqrt{13}}{(\sqrt{7} + \sqrt{6}) + \sqrt{13}} = \frac{(\sqrt{7} + \sqrt{6}) + \sqrt{13}}{{(\sqrt{7} + \sqrt{6})}^{2} - {(\sqrt{13})}^{2}} {using the identity (a - b) (a + b) = a^{2} - b^{2}} = \frac{(\sqrt{7} + \sqrt{6}) + \sqrt{13}}{(7 + 6 + 2 \times \sqrt{7} \times \sqrt{6}) - 13} = \frac{(\sqrt{7} + \sqrt{6}) + \sqrt{13}}{13 + 2 \sqrt{42} - 13} = \frac{(\sqrt{7} + \sqrt{6}) + \sqrt{13}}{2 \sqrt{42}} = \frac{(\sqrt{7} + \sqrt{6}) + \sqrt{13}}{2 \sqrt{42}} \times \frac{\sqrt{42}}{\sqrt{42}} = \frac{\sqrt{42} (\sqrt{7} + \sqrt{6}) + \sqrt{42} \times \sqrt{13}}{2 \times 42} = \frac{\sqrt{7 \times 42} + \sqrt{6 \times 42} + \sqrt{546}}{84} = \frac{\sqrt{7 \times 7 \times 6} + \sqrt{6 \times 6 \times 7} + \sqrt{546}}{84} = \frac{7 \sqrt{6} + 6 \sqrt{7} + \sqrt{546}}{84}

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