Lesson 6: Surds – Simplification and Rationalization

👨🏽‍🏫 Introduction

Have you seen numbers like or that can’t be written as exact decimals or simple fractions? These are called surds — and they help us work with square roots and irrational numbers in their exact form.

In this lesson, you’ll learn:

• What surds are and how to identify them
• How to simplify surds using square factors
• How to add, subtract, multiply, and divide surds
• How to rationalize denominators

🧠 Key Concepts

📌 1. What Is a Surd?

A surd is an irrational root that cannot be simplified into a rational number. It is left in root form for accuracy.

Examples of surds:

• (cannot be simplified further)
• Not surds: (because they simplify to rational numbers)

🧹 2. Simplifying Surds

Look for square factors inside the root and take their square root out.

Rule:

Examples:

➕➖ 3. Adding and Subtracting Surds

You can only add or subtract like surds (same root).

Examples:

• 
• 
• ❌ cannot be simplified

✖️➗ 4. Multiplying and Dividing Surds

Multiplication:

Example:

Division:

Example:

🚫 5. Rationalizing the Denominator

We prefer not to leave a surd in the denominator. To fix this, multiply top and bottom by the surd (or its conjugate).

Case 1: Single Surd Denominator

Example:

Case 2: Binomial Denominator (Use the conjugate)

Multiply by if the denominator is and vice versa.

Example:

🧮 Sample Problem Walkthroughs

1. Simplify:
   
2. Add:
3. Multiply:
4. Rationalize:

✍🏽 Practice Exercises

🅰️ A. Simplify

🅱️ B. Multiply or Divide

🅾️ C. Rationalize

✅ Answers and Explanations

• A1:
• A2:

• B1:
• B2:

• C1: Multiply by
• C2: Multiply by

📌 Recap

• ✅ Surds are roots that cannot be simplified into rational numbers
• ✅ You can only add/subtract like surds
• ✅ Use square factors to simplify surds
• ✅ Rationalize the denominator to remove surds from it

💭 Reflection Prompt

Why do you think it’s better to leave some answers in surd form instead of converting them to decimals? Where might surds appear in science or engineering?