Adding and Subtracting Radical Expressions

Combine like radicals and simplify sums and differences. Grade 11 Math 20-1.

Lesson 5.2 of Radical Expressions and Equations in Math 20-1 — Alberta curriculum lessons.

What you'll learn in this lesson

• Like Radicals. Learn what makes two radicals like terms, how the radical part acts like a variable, and why unlike radicals such as square root 2 plus square root 3 cannot be combined.
• Adding and Subtracting Like Radicals. Practice combining like radical terms directly by adding or subtracting their coefficients, grouping multiple groups of like radicals, and tracking negative signs carefully.
• Simplify First, Then Combine. Master the essential habit of fully simplifying every radical before looking for like terms — including problems with multiple radicands, cube roots, and radicals that cannot be combined after simplifying.

Inside the lesson — a free preview

What Are Like Radicals?

Definition: Like Radicals. Two radicals are called like radicals if they have both of the following:

1. The same index (both square roots, or both cube roots, etc.), AND

2. The same radicand (the same number or expression under the radical).

The coefficients can be different — that is fine. In fact, it is the coefficients that we add or subtract.

Connection to Like Terms in Algebra. Just like you can combine 2x and 3x because the variable part (x) is the same, you can combine 2√(3) and 4√(3) because the radical part (√(3)) is the same.

The radical part of a radical expression behaves exactly like a variable — it is the thing attached to the coefficient. You add the coefficients and keep the radical unchanged.

Two Important Rules

Rule 1 — Only Like Radicals Combine. You can only add or subtract radicals that are like. If they are unlike, leave them as separate terms — they cannot be simplified further.

Rule 2 — The Radicand Stays the Same. When you combine like radicals, only the coefficients add or subtract. The radicand stays unchanged — it acts like a label. You do not add the radicands.

Example: 5√(3) + 2√(3) = 7√(3), not 7√(6).

Common Mistake — Adding Radicands. False: √(2) + √(3) = √(5).

You can verify with a calculator: √(2) ≈ 1.414, √(3) ≈ 1.732, so √(2) + √(3) ≈ 3.146. But √(5) ≈ 2.236. They are not equal.

What is true: √(2) + √(2) + √(2) = 3√(2), because three copies of √(2) add up to 3 of them — just like x + x + x = 3x.

Worked example: Single Group of Like Radicals

Example 1 — Simplify 3√(5) - 7√(5). Step 1: Identify the radicals. Both have index 2 and radicand 5 — they are like.

Step 2: Subtract the coefficients. The radicand √(5) stays the same.

3√(5) - 7√(5) = (3 - 7)√(5)

Answer: -4√(5)

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