Lesson 4.8

Multiplying Radicals

Multiply radicals by multiplying coefficients together and radicands together, then simplify. For binomials, use FOIL — just like with polynomials.

Introduction

The Product Property from Lesson 4.6 works both ways: . This lesson focuses on multiplying radical expressions, including the powerful conjugate pattern that eliminates radicals entirely.

Past Knowledge

Product Property (4.6), FOIL method, difference of squares.

Today's Goal

Multiply monomial and binomial radical expressions and use conjugates.

Future Success

Conjugate multiplication is the key tool for rationalizing denominators (4.9).

Key Concepts

Monomial × Monomial

Multiply coefficients, multiply radicands, then simplify.

The Conjugate Pattern

The radical vanishes! This is the difference of squares pattern.

Worked Examples

Example 1: Simple Multiplication

Basic

Simplify .

1

Multiply coefficients and radicands

2

Simplify

Example 2: FOIL with Radicals

Intermediate

Expand .

F
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I
L

Example 3: Conjugate Pairs

Key Pattern

Multiply .

1

Apply difference of squares:

Result: — no radical! The conjugate eliminated it.

Common Pitfalls

Forgetting to Simplify

, not . Always check if the result simplifies!

Squaring a Binomial Wrong

. You must FOIL: .

Real-Life Applications

Conjugate multiplication appears in electrical engineering when working with complex impedance. The same algebraic trick is used to simplify complex fractions in AC circuit analysis.

Practice Quiz

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