<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] Use twice the conjugate:
$$\begin{aligned}
\frac{1}{1+\sqrt{2}-\sqrt{3}} &= \frac{1}{(1+\sqrt{2})-\sqrt{3}}\frac{(1+\sqrt{2})+\sqrt{3}}{(1+\sqrt{2})+\sqrt{3}}\\
&=\frac{1+\sqrt{2}+\sqrt{3}}{2 \sqrt{2}}\\
&= \frac{2+\sqrt{2}+\sqrt{6}}{4}
\end{aligned}$$
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