Lesson 4.7

Adding and Subtracting Radicals

You can only add or subtract radicals that are like radicals — same index, same radicand. Think of it like combining like terms: .

Introduction

Just as but cannot be combined, radicals follow the same rule. Sometimes you need to simplify first (using the Product Property from 4.6) before you can combine.

Past Knowledge

Combining like terms, Product Property (4.6).

Today's Goal

Add and subtract radicals by simplifying and combining like terms.

Future Success

These skills appear in multiplying radicals (4.8) and rationalizing denominators (4.9).

Key Concepts

Like Radicals

Like radicals have the same index AND the same radicand.

✓ and — like

✗ and — different radicand

✗ and — different index

The Process

1

Simplify each radical (Product Property)

2

Identify like radicals

3

Combine the coefficients

Worked Examples

Example 1: Already Like Radicals

Basic

Simplify .

1

All are — combine coefficients

Example 2: Simplify First

Intermediate

Simplify .

1

Simplify

2

Now combine like radicals

Example 3: Multiple Terms

Advanced

Simplify .

1

Simplify each

2

Combine

Common Pitfalls

Combining Unlike Radicals

. You cannot add radicands! Only like radicals can combine.

Skipping Simplification

looks impossible to combine, but after simplifying both become multiples of .

Real-Life Applications

In geometry, when you add the side lengths of a figure involving diagonal measurements (like from 45-45-90 triangles), you combine like radicals to get exact perimeters.

Practice Quiz

Loading...