Rationalize a surd $\frac{1}{1+\sqrt{2}-\sqrt{3}}$ How can I rationalize the following surd $$\frac{1}{1+\sqrt{2}-\sqrt{3}}$$ What would be the conjugate of the denominator

Rationalise twice, because after rationalising once , there would still remain a surd in the denominator.

First multiply and divide by $1+\sqrt{2}+ \sqrt{3}$

$$\frac{1}{1+\sqrt{2}-\sqrt{3}}\times\frac{1+\sqrt{2}+\sqrt{3}}{1+\sqrt{2}+\sqrt{3}}$$ $$= \frac{1+\sqrt{2}+\sqrt{3}}{1+2\sqrt{2}+2-3}=\frac{1+\sqrt{2}+\sqrt{3}}{2\sqrt{2}}$$

Now multiply and divide by $\sqrt{2}$ $$=\frac{\sqrt{2}+2+\sqrt{6}}{4}$$