A Number and a Root

Often the bottom holds a number and a root, as in $\frac{5}{3\sqrt{2}}$. Only the root is a problem: 3 is already a whole number. So multiply the top and the bottom by $\sqrt{2}$ alone:

$$\frac{5}{3\sqrt{2}} = \frac{5\sqrt{2}}{3\cdot 2} = \frac{5\sqrt{2}}{6}$$

The bottom is 3 times $\sqrt{2}\cdot \sqrt{2}$, and that product is 2, so the bottom is 6. Multiplying by the whole $3\sqrt{2}$ also clears the root, but it drags an extra 3 along: you get $\frac{15\sqrt{2}}{18}$ and then have to cancel the 3 back out.

After rationalizing, look for a common factor between the number on top and the bottom. In $\frac{7}{2\sqrt{7}}$, multiplying by $\sqrt{7}$ gives $\frac{7\sqrt{7}}{14}$, and 7 over 14 cancels to leave $\frac{\sqrt{7}}{2}$.