One Root Below

Look at $\frac{6}{\sqrt{3}}$. It is a perfectly good number, but to estimate it you would have to divide 6 by 1.732 by hand. So there is an agreement: keep roots out of the bottom of a fraction. Rewriting a fraction so its bottom has no root is called rationalizing the denominator, because the bottom becomes a rational number.

The tool is a fact from the last chapter: a root times itself is what was under it, $\sqrt{3}\cdot \sqrt{3} = 3$. Multiply the top and the bottom by $\sqrt{3}$. That is multiplying by $\frac{\sqrt{3}}{\sqrt{3}}$, which is 1, so the value does not change:

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

Now cancel the 3 into the 6, and the answer is $2\sqrt{3}$. Why bother? A whole-number bottom is easier to work with: $\frac{\sqrt{2}}{2}$ is half of 1.414, which you can do in your head, while $\frac{1}{\sqrt{2}}$ is the same number but asks you to divide by 1.414. One agreed form also makes answers easy to compare: $\frac{5}{\sqrt{5}}$ and $\sqrt{5}$ look different, and rationalizing shows they are equal.