## Rationalizing Denominators in Radical Expressions

## Introduction

The following problem is an example of simplifying square root of a fraction:

This problem seems to have been simplified as much as possible. However, square roots are *irrational numbers* and are not permitted to remain in the denominator of a fraction. Further simplification is called *rationalizing the denominator* and is required in order to change the fraction into one whose denominator is rational.

## Lesson

Everyone knows that a number divided by itself is equal to 1. The same is true of a decimal and also of a square root… the square root of a number divided by itself is equal to 1.

This is an important fact because you can multiply an answer by 1 and it remains the answer. The introduction to this lesson contained a fractional answer that contained an irrational number in the denominator. You aren’t allowed to leave your final answer in this format, so rationalize the denominator as follows:

Rational numbers include all positive and negative whole numbers as well as all terminating decimals. When dealing with fractions, the denominator will generally be a whole number after rationalizing the denominator (only in an unusual case will the denominator be a decimal after it has been rationalized)

**Example 1****:** Simplify the fraction

Solution:

Some problems result in an answer that has a radical in the denominator. In this case, you should automatically continue simplifying by rationalizing the denominator.

**Example 2****:** Simplify the fraction

Solution:

**Example 3****:** Simplify the fraction

Solution:

When simplifying radical expressions, remember to begin by finding the largest possible perfect square number that is a factor of the number in the radical. In example 2 (for example), 4 is the perfect square factor of 28 and 25 is the largest perfect square factor of 75. Once you have identified the largest perfect square factor, simply use the properties of multiplication, division, and square roots to find the final answer.