<s>[INST] Write $\frac{\sqrt{2}}{\sqrt{2} + \sqrt{3} - \sqrt{5}} - \frac{1}{2}$ with a rational denominator. Write $\frac{\sqrt{2}}{\sqrt{2} + \sqrt{3} - \sqrt{5}} - \frac{1}{2}$ with a rational denominator.
How should I solve this question?
 [/INST] Only the first term
We have $\frac{\sqrt 2}{\sqrt 2 + \sqrt 3 - \sqrt 5}.$ Do :
$$
\frac{\sqrt 2}{\sqrt 2 + \sqrt 3 - \sqrt 5}= \frac{\sqrt 2(\sqrt 2 + \sqrt 3  + \sqrt 5)}{(\sqrt 2 + \sqrt 3)^2 - 5} = \frac{2 + \sqrt 6 + \sqrt {10}}{2\sqrt 6} \\
= \frac{2\sqrt 6 + 6 + \sqrt {60}}{2\sqrt 6 \sqrt 6} = \frac{6 + 2 \sqrt 6 + \sqrt {60}}{12}= \frac{3+\sqrt 6 + \sqrt {15}}{6}
$$
So you are correct!
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