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

How do you add and subtract radical expressions?

You add and subtract radical expressions by combining like radicals, which have the same index and the same radicand. Treat the radical like a variable and combine only the numbers in front. For example, 3√5 + 7√5 = 10√5, and 9√2 − 4√2 = 5√2. Terms such as √5 and √2 are not like radicals, so 3√5 + 4√2 stays as it is.

How do you add and subtract radical expressions with different square roots?

Simplify each square root first, because different-looking radicals often turn into like radicals. Pull out the largest perfect-square factor of each radicand. For √12 + √27, write √12 = √(4 × 3) = 2√3 and √27 = √(9 × 3) = 3√3. Now the terms match, so 2√3 + 3√3 = 5√3. If the radicands still differ after simplifying, the terms cannot be combined.

How do you add and subtract radical expressions with variables?

Radicals with variables combine the same way: simplify, then add or subtract the coefficients of matching radicals. For 4√x + 2√x − √x, the result is 5√x. A harder one is √(18x) + √(8x). Simplify to 3√(2x) + 2√(2x), then combine to get 5√(2x). In real-number problems, assume x ≥ 0 so the square roots are defined.

How do you add, subtract and multiply radical expressions?

Adding and subtracting need like radicals, but multiplying does not. To multiply, use √a × √b = √(ab): √2 × √8 = √16 = 4. To add the same two numbers, simplify first: √8 = 2√2, so √2 + √8 = √2 + 2√2 = 3√2. Mixing the rules up, for example writing √2 + √8 = √10, is one of the most common mistakes.

How do you add and subtract radical expressions with fractions?

Find a common denominator just as you would with ordinary fractions, then combine like radicals in the numerator. For √3/2 + √3/3, use 6 as the denominator: 3√3/6 + 2√3/6 = 5√3/6. If a radical sits in a denominator, rationalize it first. For 1/√2 + √2, rewrite 1/√2 as √2/2, then √2/2 + 2√2/2 = 3√2/2.

Related Topics

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