Lesson 66: Multiplying Radicals

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• Lesson Description
• Learning Objectives
• Topics Covered
• Key Terms
• Lesson Examples
• Practice Problems
• Tips & Common Mistakes
• Summary
• Challenge Problems
• Unit Navigation

Lesson Description

In this lesson, students learn how to multiply radical expressions using the product property of square roots. We will focus on multiplying numbers under radicals, simplifying results, and applying the distributive property when multiplying binomials containing radicals.

Learning Objectives

• Apply the product property of radicals: √a · √b = √(ab).
• Multiply radicals and simplify the result fully.
• Use the distributive property when multiplying expressions containing radicals.

Topics Covered

• Product property of square roots
• Simplifying radicals after multiplication
• Multiplying binomials with radicals (FOIL method)

Key Terms

• Radical: An expression that includes a square root or other root symbol.
• Product Property of Radicals: √a · √b = √(ab), for a ≥ 0 and b ≥ 0.
• Simplest Radical Form: A radical expression with no perfect square factors inside the radical.

Lesson Examples

Example 1: Multiply Two Radicals

1. Multiply inside the radical: √3 · √12 = √(3 × 12).
2. Simplify inside: √36.
3. Take the square root: √36 = 6.

Example 2: Multiply and Simplify

1. Multiply: √5 · √20 = √(5 × 20).
2. Simplify inside: √100.
3. Take the square root: √100 = 10.

Example 3: Multiply Binomials with Radicals

1. Multiply (√2 + 3)(√2 − 3).
2. Use FOIL: (√2)(√2) − 3√2 + 3√2 − 9.
3. Simplify: 2 − 9 = −7.

Practice Problems

1. √4 · √9
2. √6 · √24
3. (√3 + 2)(√3 − 2)
4. √8 · √2

Tips & Common Mistakes

• Always multiply inside the radical first before simplifying.
• Look for perfect square factors when simplifying.
• Be careful when using FOIL—combine like terms properly.

Summary

Multiplying radicals relies on the product property: √a · √b = √(ab). After multiplying, simplify completely by factoring out perfect squares. When multiplying expressions with multiple terms, use the distributive property and simplify all results.

Challenge Problems

1. √12 · √18
2. (2√5)(3√10)
3. (√7 + √3)(√7 − √3)