Last time we learned about properties of square roots. Let's take a short review.

Find the simplified form of the given radical expression.

After selecting your answer, click the “Submit” button.

Now, let's move on with today's lesson.

In the previous lesson, we simplified some radical expressions using the product and quotient properties of square roots.

Let’s look at some general rules for getting a radical expression into simplest form.

First, a radicand must not have a perfect square factor other than 1.

Second, a radicand may not contain a fraction.

Third, a radical may not appear in the denominator of a fraction.

Now, let’s look at an example of the first rule. Here we have the radical expression: <EQUATION>

We begin by finding the prime factorization of the radicand.

This gives us: <EQUATION>

Next, we use the product property of square roots to rewrite the expression. We place the perfect squares <EQUATION> and <EQUATION> under their own radical symbols.

We are left with a third radical term: <EQUATION>

What is the final simplified form of <EQUATION>? Click the “Submit” button after selecting your answer.

We can simplify the perfect squares. This gives us: <EQUATION>

The number 2 is not a perfect square. Therefore, the expression is in its simplest form.

Here we have the radical expression: <EQUATION>

This violates our second rule because the radicand contains a fraction.

Let’s rewrite the expression using the quotient property of square roots. This gives: <EQUATION>

The numerator further simplifies to 9.

We no longer have a radicand with a fraction. However, we have <EQUATION> in the denominator of a fraction. This violates rule three.

We can eliminate the radical in the denominator by multiplying the numerator and denominator by <EQUATION>.

Now, what is the simplified form of <EQUATION>? Click the “Submit” button after selecting your answer.

We find that: &space; &space; &space; &space; &space; &space; &space; &space; &space; &space; &space; &space; <EQUATION><EQUATION>

The method we used to remove the radical from the denominator is called rationalizing the denominator.

Given a radical expression of the form: <EQUATION>

We rationalize the denominator by multiplying the numerator and denominator by <EQUATION>.

This gives the expression: &space; &space; &space; &space; <EQUATION>

It is required that: <EQUATION>and<EQUATION>

Now, you try some examples.

Simplify the given radical expression.

After selecting your answer, click the “Submit” button.

In the next lesson, we will learn about operations with radical expressions.

Copyright 2006 The Regents of the University of California and Monterey Institute for Technology and Education