\[\begin{equation} \begin{aligned} \frac{1}{1-\sqrt{3}-\sqrt{7}}&=\frac{1}{1-\sqrt{3}-\sqrt{7}} \times \frac{1+\sqrt{3}+\sqrt{7}}{1+\sqrt{3}+\sqrt{7}}\\ &=\frac{1+\sqrt{3}+\sqrt{7}}{1-(\sqrt{3}+\sqrt{7})^2}\\ &= \frac{1+\sqrt{3}+\sqrt{7}}{-9-2 \sqrt{21}} \\ &= \frac{1+\sqrt{3}+\sqrt{7}}{-9-2 \sqrt{21}} \times \frac{-9+2 \sqrt{21}}{-9+2 \sqrt{21}} \\ &=- \frac{(1-\sqrt{3}-\sqrt{7})(-9+2\sqrt{21})}{3} \end{aligned} \end{equation}\]
Rationalising a Denominator With Two Surds
For a fraction we two surd terms in the denominator, we can rationalise just as for a single surd, by multiplying by a fraction with a sign change for the surd(s) in the denominator.