Open Resources for Community College Algebra

Subsection 6.2.1 Rationalizing the Denominator

To simplify radical expressions, we have seen that it helps to make the radicand as small as possible. Another helpful principle is to not leave any irrational numbers, such as \(\sqrt{3}\) or \(2\sqrt{5}\text{,}\) in the denominator of a fraction. In other words, we want the denominator to be rational. The process of dealing with such numbers in the denominator is called rationalizing the denominator.

Let's see how we can replace \(\frac{1}{\sqrt{5}}\) with an equivalent expression that has no radical expressions in its denominator. If we multiply a radical by itself, the result is the radicand, by Definition 6.1.2. As an example:

With \(\frac{1}{\sqrt{5}}\text{,}\) we may multiply both the numerator and denominator by the same non-zero number and have an equivalent expression. If we multiply the numerator and denominator by \(\sqrt{5}\text{,}\) we have:

And voilà, we have an expression with no radical in its denominator. We can use a calculator to verify that \(\frac{1}{\sqrt{5}}\approx0.4472\text{,}\) and also \(\frac{\sqrt{5}}{5}\approx0.4472\text{.}\) They are equal.

Subsection 6.2.2 Rationalize the Denominator Using the Difference of Squares Formula

Conside the number \(\frac{1}{\sqrt{2}+1}\text{.}\) Its denominator is irrational, approximately \(2.414\ldots\text{.}\) Can we rewrite this as an equivalent expression where the denominator is rational? Let's try multiplying the numerator and denominator by \(\sqrt{2}\text{:}\)

We removed one radical from the denominator, but created another. We need to find another method. The difference of squares formula will help:

Those two squares in \(a^2-b^2\) can be used as a tool to annihilate radicals. Take \(\frac{1}{\sqrt{2}+1}\text{,}\) and multiply both the numerator and denominator by \(\sqrt{2}-1\text{:}\)