Add and subtract radical fractions: different denominators

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Learn it Different denominators: rationalize each fraction on its own, then add. If the denominators are conjugates, their product is a whole number, so use it as the common denominator.

Example: $\frac{5\sqrt{18}}{2 - 3\sqrt{2}} - \frac{3\sqrt{18}}{2 + 3\sqrt{2}}$

$\sqrt{18} = 3\sqrt{2}$, so the tops are $15\sqrt{2}$ and $9\sqrt{2}$.
Common denominator: $(2 - 3\sqrt{2})(2 + 3\sqrt{2}) = 4 - 18 = -14$.
Top: $15\sqrt{2}(2 + 3\sqrt{2}) - 9\sqrt{2}(2 - 3\sqrt{2}) = (30\sqrt{2} + 90) - (18\sqrt{2} - 54) = 144 + 12\sqrt{2}$.
$\frac{144 + 12\sqrt{2}}{-14} = -\frac{72 + 6\sqrt{2}}{7}$.

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