Lesson 4.9

Rationalizing the Denominator

A simplified radical expression should never have a radical in the denominator. We eliminate it by multiplying top and bottom by the right expression — a process called rationalizing.

Introduction

Writing is technically correct, but convention demands a rational denominator. Multiply by to get . For binomial denominators, use the conjugate from Lesson 4.8.

Past Knowledge

Product Property (4.6), conjugate multiplication (4.8), equivalent fractions.

Today's Goal

Rationalize denominators with monomial and binomial radical expressions.

Future Success

Rationalizing is essential for solving radical equations and simplifying rational exponents.

Key Concepts

Monomial Denominator

Multiply by :

Binomial Denominator

Multiply by the conjugate:

Worked Examples

Example 1: Monomial Denominator

Basic

Rationalize .

1

Multiply by

Example 2: Coefficient in Denominator

Intermediate

Rationalize .

1

Multiply by

Example 3: Binomial Denominator (Conjugate)

Advanced

Rationalize .

1

Multiply by the conjugate

2

Simplify denominator (difference of squares)

Common Pitfalls

Multiplying by the Same Thing

For , multiply by (conjugate), NOT again. The latter gives but also cross terms.

Forgetting to Reduce

After rationalizing, always check if you can simplify the resulting fraction (like ).

Real-Life Applications

In physics, exact answers like (the sine of 45°) are universally preferred over . Rationalized forms are the standard in science and engineering.

Practice Quiz

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