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Adding surd fractions

Fractions with surds can be added or subtracted as regular fractions. However, the denominator should be rationalised first.

$$$ \frac{1}{\Tblue{\sqrt{2}}} + \frac{1}{\Tgreen{\sqrt{3}}} = \frac{\Tblue{\sqrt{2}}}{2} + \frac{\Tgreen{\sqrt{3}}}{3} =\frac{3\sqrt{2}+2\sqrt{3}}{6} $$$

Here is another example.

\begin{align*} \frac{\sqrt{2}}{\Tblue{1+\sqrt{3}}} - \frac{5}{\Tgreen{\sqrt{2}}} &=\frac{\sqrt{2}(\Tblue{1-\sqrt{3}})}{1-3} - \frac{5\Tgreen{\sqrt{2}}}{2} \\ &=\frac{\sqrt{2}(\sqrt{3}-1)}{2} - \frac{5\sqrt{2}}{2} \\ &=\frac{\sqrt{2}(\sqrt{3}-1-5)}{2} \\ &=\frac{\sqrt{2}(\sqrt{3}-6)}{2} \\ &=\frac{\sqrt{6}}{2} -3\sqrt{2} \end{align*}