Rationalizing the Denominator

When you see a fraction like $\frac{6}{\sqrt{2}}$ or $\frac{7}{\sqrt{5}}$, the denominator contains a radical (a square root). Mathematicians prefer to rewrite these fractions so that the denominator is a whole number — this is called rationalizing the denominator.

There are two main situations you'll encounter:

1. A simple radical in the denominator: $\frac{b}{\sqrt{a}}$
2. A binomial with a radical in the denominator: $\frac{c}{a+\sqrt{b}}$

Each situation requires a slightly different technique, both of which you'll master in this subtopic.

Consider the fraction $\frac{9}{\sqrt{3}}$. How would you actually calculate this?

Dividing by $\sqrt{3}\approx \mathrm{1.732050808...}$ (an irrational number with infinite, non-repeating decimal places) is extremely messy. But if we rewrite the fraction as $\frac{9\sqrt{3}}{3}=3\sqrt{3}$, we only need to divide by 3 — a whole number.

Rationalizing also makes it much easier to compare two expressions. For example, which is bigger: $\frac{5}{\sqrt{3}}$ or $\frac{4}{\sqrt{2}}$?

After rationalizing:

• $\frac{5}{\sqrt{3}}=\frac{5\sqrt{3}}{3}\approx \frac{5\times 1.732}{3}\approx \frac{8.660}{3}\approx 2.89$
• $\frac{4}{\sqrt{2}}=\frac{4\sqrt{2}}{2}=2\sqrt{2}\approx 2\times 1.414\approx 2.83$

Now comparison is simple: $\frac{5}{\sqrt{3}}>\frac{4}{\sqrt{2}}$.