One Root Below

Look at 6/√(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, √(3) × √(3) = 3. Multiply the top and the bottom by √(3). That is multiplying by √(3)/√(3), which is 1, so the value does not change:

6/√(3) = (6√(3))/(√(3) × √(3)) = (6√(3))/3

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