Lesson 68: Radical Expressions

Page Menu

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

Lesson Description

Radical expressions include square roots (and sometimes higher roots) that contain numbers, variables, or both. In this lesson, students learn how to simplify radical expressions, combine like radicals, and evaluate them when variable values are provided.

Learning Objectives

• Simplify radical expressions containing numbers and variables.
• Combine like radical terms correctly.
• Evaluate radical expressions by substituting given values.

Topics Covered

• Simplifying radicals with variables
• Combining like radicals
• Evaluating radical expressions

Key Terms

• Radical Expression: An expression that contains a radical symbol.
• Like Radicals: Radicals that have the same index and radicand.
• Evaluate: To substitute a value for a variable and simplify.

Lesson Examples

Example 1: Simplifying a Radical with Variables

1. Simplify √(16x²).
2. Break apart: √16 · √x².
3. Simplify: 4x (assuming x ≥ 0).

Example 2: Combining Like Radicals

1. Simplify 3√5 + 2√5.
2. Since both have √5, combine coefficients.
3. Answer: 5√5.

Example 3: Evaluating a Radical Expression

1. Evaluate √(9a) when a = 4.
2. Substitute: √(9 × 4) = √36.
3. Answer: 6.

Practice Problems

1. Simplify √(25y²).
2. Combine: 4√3 + 6√3.
3. Evaluate √(4b) when b = 9.
4. Simplify √(18x²).

Tips & Common Mistakes

• Only combine radicals that have the same radicand.
• Always factor out perfect squares when simplifying.
• Remember to substitute values before taking the square root when evaluating.

Summary

Radical expressions can include numbers and variables. To simplify, factor out perfect squares and reduce to simplest form. Like radicals can be combined by adding or subtracting coefficients. When evaluating, substitute first, then simplify.

Challenge Problems

1. Simplify 2√12 + 3√27.
2. Evaluate √(8m²) when m = 5.
3. Simplify (√2 + 3√2) − √2.