Lesson 4.6

Product Property of Radicals

The Product Property lets you split a radical into factors: . This is the key to simplifying non-perfect roots.

Introduction

Not every radicand is a perfect square. When you encounter , you simplify it by finding a perfect-square factor inside: .

Past Knowledge

Nth roots (4.5), perfect squares and cubes, prime factorization.

Today's Goal

Use the Product Property to simplify radicals with numeric and variable radicands.

Future Success

Simplified radicals are needed for adding/subtracting (4.7) and multiplying radicals (4.8).

Key Concepts

The Product Property

Works for any index n, as long as the roots are real.

Strategy: Find the largest perfect nth-power factor.

For : largest perfect square factor of 72 is 36.

Perfect Powers Reference

Perfect Squares

4, 9, 16, 25, 36, 49, 64, 81, 100

Perfect Cubes

8, 27, 64, 125, 216

Variables:

Divide the exponent by the index.

Worked Examples

Example 1: Numeric Square Root

Basic

Simplify .

1

Find largest perfect square factor

2

Apply Product Property

Example 2: Cube Root

Intermediate

Simplify .

1

Find largest perfect cube factor

2

Apply Product Property

Example 3: Variables

Advanced

Simplify .

1

Break into perfect and non-perfect parts

2

Simplify perfect squares out

Common Pitfalls

Not Using the Largest Factor

is technically correct but not fully simplified. Always use the largest perfect factor (36).

Adding Inside vs. Multiplying

. The Product Property is for multiplication only!

Real-Life Applications

Engineers simplify radicals when calculating distances, areas, and forces. A simplified form like is cleaner and more useful in formulas than .

Practice Quiz

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