- The Division Property of Square Roots
- Rationalizing the Denominator
- Conjugates and Rationalizing the Denominator

Summary: This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. The distinction between the principal square root of the number x and the secondary square root of the number x is made by explanation and by example. The simplification of the radical expressions that both involve and do not involve fractions is shown in many detailed examples; this is followed by an explanation of how and why radicals are eliminated from the denominator of a radical expression. Real-life applications of radical equations have been included, such as problems involving daily output, daily sales, electronic resonance frequency, and kinetic energy. Objectives of this module: be able to use the division property of square roots, the method of rationalizing the denominator, and conjugates to divide square roots.

- The Division Property of Square Roots
- Rationalizing the Denominator
- Conjugates and Rationalizing the Denominator

In our work with simplifying square root expressions, we noted that

Since this is an equation, we may write it as

To divide two square root expressions, we use the division property of square roots.

The quotient of the square roots is the square root of the quotient.

As we can see by observing the right side of the equation governing the division of square roots, the process may produce a fraction in the radicand. This means, of course, that the square root expression is not in simplified form. It is sometimes more useful to rationalize the denominator of a square root expression before actually performing the division.

Simplify the square root expressions.

This radical expression is not in simplified form since there is a fraction under the radical sign. We can eliminate this problem using the division property of square roots.

A direct application of the rule produces

The rule produces the quotient quickly. We could also rationalize the denominator first and produce the same result.

Some observation shows that a direct division of the radicands will produce a fraction. This suggests that we rationalize the denominator first.

Simplify the square root expressions.

To perform a division that contains a binomial in the denominator, such as *conjugate* of the denominator.

A conjugate of the binomial

Notice that when the conjugates

This principle helps us eliminate square root radicals, as shown in these examples that illustrate finding the product of conjugates.

Simplify the following expressions.

The conjugate of the denominator is

The conjugate of the denominator is

Simplify the following expressions.

For the following problems, simplify each expressions.

2

6

3

0

*((Reference))* Simplify

*((Reference))* Solve the compound inequality

*((Reference))* Construct the graph of

*((Reference))* The symbol

*((Reference))* Simplify

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