Rationalizing the Denominator

A root in the denominator

A fraction such as $\frac{1}{\sqrt{2}}$ has a root in its denominator. Its value is fine, but the form is awkward. To find it as a decimal, you would divide 1 by 1.41421…, a decimal that never ends. And to add $\frac{1}{\sqrt{2}}$ to another fraction, you would need a common denominator that holds a root.

So a fraction like this is rewritten with a whole number in the denominator. The new denominator is a rational number, so the rewriting is called rationalizing the denominator.

Multiply by 1

Multiplying a fraction by 1 never changes its value, and $\frac{\sqrt{2}}{\sqrt{2}}$ is 1, because any number other than 0 divided by itself is 1. So multiply both the numerator and the denominator of $\frac{1}{\sqrt{2}}$ by $\sqrt{2}$, which is $\frac{1}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}}$. The numerator becomes $1 \times \sqrt{2} = \sqrt{2}$, and the denominator becomes $\sqrt{2} \times \sqrt{2}$.

In the denominator, $\sqrt{2} \times \sqrt{2}$ is exactly 2, because $\sqrt{2}$ is the number whose square is 2. So the root is gone from the denominator: $\frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$.

The new form is easy to work with. $\frac{\sqrt{2}}{2}$ is half of $\sqrt{2}$, which is about 1.414 ÷ 2 = 0.707.

$\frac{1}{\sqrt{2}}$ and $\frac{\sqrt{2}}{2}$ are the same point on the number line, a little past 0.7. Only the way the number is written has changed.

Then simplify

The same move works with any single root in the denominator. For $\frac{6}{\sqrt{3}}$, multiply the numerator and the denominator by $\sqrt{3}$. The numerator becomes $6 \times \sqrt{3} = 6\sqrt{3}$, and the denominator becomes $\sqrt{3} \times \sqrt{3} = 3$, so $\frac{6}{\sqrt{3}} = \frac{6\sqrt{3}}{3}$.

Now the denominator is 3, and 6 ÷ 3 = 2, so $\frac{6\sqrt{3}}{3} = 2\sqrt{3}$. Check it by multiplying back: $2\sqrt{3} \times \sqrt{3} = 2 \times 3 = 6$, so $6 \div \sqrt{3}$ really is $2\sqrt{3}$.

For $\frac{10}{\sqrt{5}}$, multiply the numerator and the denominator by $\sqrt{5}$. The denominator becomes $\sqrt{5} \times \sqrt{5} = 5$, so $\frac{10}{\sqrt{5}} = \frac{10\sqrt{5}}{5}$, and 10 ÷ 5 = 2 gives $2\sqrt{5}$.

Common mistakes

Multiply the numerator and the denominator by the same root. Multiplying only the denominator of $\frac{1}{\sqrt{2}}$ by $\sqrt{2}$ gives $\frac{1}{2}$, which is a different number: 0.5, not 0.707.

Keep the root after dividing. In $\frac{6\sqrt{3}}{3}$, the 3 in the denominator divides the 6, and the $\sqrt{3}$ stays in the numerator: the answer is $2\sqrt{3}$, not 2 and not $6\sqrt{3}$.