<s>[INST] 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
 [/INST] 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 $\sqrt{2}$
$$=\frac{\sqrt{2}+2+\sqrt{6}}{4}$$
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