Rationalizing with the conjugate

Statement

Rationalize the denominator of $\dfrac{6}{\sqrt{10} - \sqrt{6}}$.

Solution

Multiply numerator and denominator by the conjugate $\sqrt{10} + \sqrt{6}$, to get a difference of squares in the denominator: $\frac{6}{\sqrt{10} - \sqrt{6}} \cdot \frac{\sqrt{10} + \sqrt{6}}{\sqrt{10} + \sqrt{6}} = \frac{6(\sqrt{10} + \sqrt{6})}{(\sqrt{10})^2 - (\sqrt{6})^2}.$ The denominator becomes $10 - 6 = 4$, hence $\frac{6(\sqrt{10} + \sqrt{6})}{4} = \frac{3(\sqrt{10} + \sqrt{6})}{2}.$ $\boxed{\dfrac{3(\sqrt{10} + \sqrt{6})}{2}}$