## Rationalizing a Denominator

This post about rationalizing a denominator is part of a series of posts to help you prepare for the Advanced Algebra and Functions part of the Accuplacer test.

## Question

Consider the expression

$\fn_jvn {\color{Blue} \frac{a-b}{\sqrt{a}-\sqrt{b}} }$

where a and b are positive integers.

Which of the following expressions is equivalent to it?

A. $\fn_jvn \frac{a-b}{\sqrt{a-b}}$

B. $\fn_jvn \sqrt{a-b}$

C. $\fn_jvn \sqrt{a}+\sqrt{b}$

D. $\fn_jvn a\sqrt{a}+b\sqrt{b}$

## Solution

Only one of the four answer choices is a fraction. See if you can convert the expression that you are given into a format that is not a fraction and perhaps get one of the answer choices. Is there a way to make that denominator go away?

### Sidebar: Rationalizing a denominator using a conjugate

A denominator that contains radicals is considered poor form. You may sometimes be asked to rationalize such denominators, meaning convert the fraction or rational expression into a format that has no radicals in its denominator. The usual way to do that is to use the denominator’s conjugate. The conjugate of an expression $\inline \fn_jvn c+d$ is $\inline \fn_jvn c-d$.

For $\inline \fn_jvn \frac{a-b}{\sqrt{a}-\sqrt{b}}$, multiply and divide by the denominator’s conjugate to rationalize the denominator, which is likely a step in the right direction. (To multiply an expression by something and divide by the same thing changes the expression’s form but not its value.) Let’s try that.

The conjugate of

$\fn_jvn {\color{Blue} \sqrt{a}-\sqrt{b}}$

is

$\fn_jvn {\color{Blue} \sqrt{a}+\sqrt{b}}$

Multiplying and dividing $\inline \fn_jvn \frac{a-b}{\sqrt{a}-\sqrt{b}}$ by the conjugate looks like:

$\fn_jvn {\color{Blue} \frac{a-b}{\sqrt{a}-\sqrt{b}} \cdot \frac {\sqrt{a}+\sqrt{b}}{\sqrt{a}+\sqrt{b}}=\frac{\left ( a-b \right )\left ( \sqrt{a}+\sqrt{b} \right )}{a-b}}$

All the radicals have dropped out of the denominator. That’s one thing we were looking for. Still, we can simplify: Notice that $\inline \fn_jvn a-b$ is a factor in both the numerator and the denominator. Cancel it and you get

$\fn_jvn {\color{Blue} \frac{\left ( a-b \right )\left ( \sqrt{a}+\sqrt{b} \right )}{a-b}=\sqrt{a}+\sqrt{b}}$

This is answer choice C. So you’re done.

For more about rational numbers, expressions, and equations, see Solving a Rational Equation

The Accuplacer sample problem on which this one is based is #19 in the sample questions for the Accuplacer Advanced Algebra and Functions test.