Rationalizing the Denominator: Taming Surds

When working with surds, you might come across fractions with square roots in the denominator. To simplify these, we use a process called rationalizing the denominator. It sounds fancy, but it just means getting rid of the surd in the denominator to make the fraction easier to work with.

What Does “Rationalizing the Denominator” Mean?

Rationalizing the denominator means rewriting a fraction so there are no square roots (or surds) in the denominator. This makes calculations and comparisons easier.

For example, instead of having $\frac{1}{\sqrt{2}}$, we can rewrite it without the surd in the denominator.

How Do We Rationalize the Denominator?

To get rid of the surd in the denominator, we multiply the fraction by a special version of 1 that removes the surd from the denominator.

Here’s how it works:

1. Identify the surd in the denominator.
2. Multiply the fraction by that surd over itself (this is like multiplying by 1, so it doesn’t change the value of the fraction).
3. Simplify the expression if possible.

Example: Rationalizing $\frac{1}{\sqrt{2}}$

1. Multiply by $\frac{\sqrt{2}}{\sqrt{2}}$: $$\frac{1}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2}$$

Now, $\frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$, with no surd in the denominator

 

 

 

Rationalizing with Two Terms in the Denominator

If there are two terms in the denominator (like $1 + \sqrt{3}$), we multiply by the conjugate of the denominator.

The conjugate is the same expression but with the opposite sign in the middle. For example, the conjugate of $1 + \sqrt{3}$ is $1 - \sqrt{3}$.

Why Use the Conjugate?

Using the conjugate removes the surd by creating a difference of squares, which simplifies to a whole number.

Example: Rationalizing $\frac{3}{1 + \sqrt{3}}$

1. Multiply by the conjugate $\frac{1 - \sqrt{3}}{1 - \sqrt{3}}$: $$\frac{3}{1 + \sqrt{3}} \times \frac{1 - \sqrt{3}}{1 - \sqrt{3}} = \frac{3(1 - \sqrt{3})}{(1 + \sqrt{3})(1 - \sqrt{3})}$$
2. Expand the numerator: $$= \frac{3 - 3\sqrt{3}}{(1 + \sqrt{3})(1 - \sqrt{3})}$$
3. Simplify the denominator using the difference of squares formula: $$= \frac{3 - 3\sqrt{3}}{1 - 3} = \frac{3 - 3\sqrt{3}}{-2} = -\frac{3 - 3\sqrt{3}}{2}$$

So, $\frac{3}{1 + \sqrt{3}} = -\frac{3 - 3\sqrt{3}}{2}$

 

Tips for Rationalizing the Denominator

• Single Surd: Multiply by that surd over itself (like $\frac{\sqrt{2}}{\sqrt{2}}$
• Two Terms (e.g., $a + \sqrt{b}$): Multiply by the conjugate, which is $a - \sqrt{b}$