How to Add and Subtract Radical Expressions? (+FREE Worksheet!)

Adding and subtracting radical expressions works just like combining like terms with variables — you can only combine radicals that have the same index and the same radicand. Mastering this skill requires simplifying each radical first, then grouping and combining the ones that match. This guide shows you the complete process with worked examples, two video lessons, and practice problems.

What Are Like Radicals?

Like radicals (also called similar radicals) have the same index and the same radicand. Just as \(\color{blue}{3x + 5x = 8x}\), you can combine like radicals by adding or subtracting their coefficients:

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• \(\color{blue}{3\sqrt{2} + 5\sqrt{2} = 8\sqrt{2}}\)
• \(\color{blue}{4\sqrt{3} – \sqrt{3} = 3\sqrt{3}}\)

Unlike radicals — for example, \(\color{blue}{\sqrt{2}}\) and \(\color{blue}{\sqrt{3}}\) — cannot be combined.

How to Add and Subtract Radical Expressions

1. Simplify Each Radical First

Before trying to combine radicals, simplify each one completely. Radicals that look different may become like radicals after simplification.

• \(\color{blue}{\sqrt{12} = 2\sqrt{3}}\) and \(\color{blue}{\sqrt{27} = 3\sqrt{3}}\)
• Now both have \(\color{blue}{\sqrt{3}}\): \(\color{blue}{2\sqrt{3} + 3\sqrt{3} = 5\sqrt{3}}\)

2. Identify Like Radicals

Group terms that have the same radicand. Ignore coefficients when determining whether radicals are “like.”

3. Combine by Adding or Subtracting Coefficients

Add or subtract only the coefficients; keep the radical part unchanged.

Step-by-Step Summary

1. Simplify every radical in the expression.
2. Identify all pairs of like radicals (same radicand).
3. Add or subtract the coefficients of like radicals.
4. Leave unlike radicals as separate terms.

Watch: Adding & Subtracting Radical Expressions (Video Lesson)

The Organic Chemistry Tutor covers square roots and cube roots with thorough step-by-step examples:

Adding and Subtracting Radical Expressions — Worked Examples

Example 1: Add \(\color{blue}{3\sqrt{2} + 5\sqrt{2}}\).

Like radicals (both \(\color{blue}{\sqrt{2}}\)). Add coefficients: \(\color{blue}{3 + 5 = 8}\).
Answer: \(\color{blue}{8\sqrt{2}}\)

Example 2: Subtract \(\color{blue}{4\sqrt{3} – \sqrt{3}}\).

Like radicals. Subtract: \(\color{blue}{4 – 1 = 3}\).
Answer: \(\color{blue}{3\sqrt{3}}\)

Example 3: Add \(\color{blue}{\sqrt{12} + \sqrt{27}}\).

Simplify: \(\color{blue}{\sqrt{12} = 2\sqrt{3}}\); \(\color{blue}{\sqrt{27} = 3\sqrt{3}}\). Now like radicals: \(\color{blue}{2\sqrt{3} + 3\sqrt{3} = 5\sqrt{3}}\).

Example 4: Simplify \(\color{blue}{2\sqrt{8} – \sqrt{32}}\).

Simplify: \(\color{blue}{2\sqrt{8} = 2 \times 2\sqrt{2} = 4\sqrt{2}}\); \(\color{blue}{\sqrt{32} = 4\sqrt{2}}\). Subtract: \(\color{blue}{4\sqrt{2} – 4\sqrt{2} = 0}\).

More Practice: Video Walkthrough

This video focuses specifically on square root terms with step-by-step coefficient work:

Exercises for Adding and Subtracting Radical Expressions

Simplify each expression.

1. \(\color{blue}{2\sqrt{5} + 7\sqrt{5}}\)
2. \(\color{blue}{6\sqrt{3} – 2\sqrt{3}}\)
3. \(\color{blue}{\sqrt{18} + \sqrt{50}}\)
4. \(\color{blue}{3\sqrt{12} – \sqrt{48}}\)
5. \(\color{blue}{2\sqrt{45} + \sqrt{20}}\)
6. \(\color{blue}{4\sqrt{7} – 2\sqrt{7} + \sqrt{7}}\)

Answers

1. \(\color{blue}{9\sqrt{5}}\)
2. \(\color{blue}{4\sqrt{3}}\)
3. \(\color{blue}{8\sqrt{2}}\)  (since \(\color{blue}{\sqrt{18} = 3\sqrt{2}}\) and \(\color{blue}{\sqrt{50} = 5\sqrt{2}}\))
4. \(\color{blue}{2\sqrt{3}}\)  (since \(\color{blue}{3\sqrt{12} = 6\sqrt{3}}\) and \(\color{blue}{\sqrt{48} = 4\sqrt{3}}\))
5. \(\color{blue}{8\sqrt{5}}\)  (since \(\color{blue}{2\sqrt{45} = 6\sqrt{5}}\) and \(\color{blue}{\sqrt{20} = 2\sqrt{5}}\))
6. \(\color{blue}{3\sqrt{7}}\)

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Want More Practice?

We haven’t published a worksheet built specifically for Adding and Subtracting Radical Expressions just yet. In the meantime, the free worksheets below cover closely related skills and concepts. If you’d like extra practice, download any that look helpful, complete the problems, and check your work — they’re a great way to reinforce what you learned on this page and strengthen the foundations this topic builds on:

• Download Adding and Subtracting Polynomials Worksheet
• Download Graphing Square Root Functions Worksheet

Frequently Asked Questions

Why must you simplify radicals before combining them?

Two radicals may look unlike but become like radicals once simplified. For example, \(\color{blue}{\sqrt{12}}\) and \(\color{blue}{\sqrt{27}}\) both simplify to multiples of \(\color{blue}{\sqrt{3}}\), so they can be combined. Without simplifying first, you would incorrectly leave them as separate terms.

Can you add √2 and √3?

No. They are unlike radicals because their radicands are different. The expression \(\color{blue}{\sqrt{2} + \sqrt{3}}\) is already in simplest form and cannot be combined further.

What if the coefficients of like radicals add up to zero?

The result is zero. For example, \(\color{blue}{4\sqrt{2} – 4\sqrt{2} = 0}\). The radical terms cancel completely.

Related Topics

• Simplifying Radical Expressions
• Multiplying Radical Expressions
• Rationalizing Radical Expressions
• Radical Equations
• Domain and Range of Radical Functions