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# Adding and subtracting surds
- URL: https://www.esheets.io/adding-and-subtracting-surds/
- Published: 2025-03-27T12:57:18.000Z
- Updated: 2026-06-21T16:35:20.000Z
- Author: Richard Linnington
- Tags: Maths, Number

Surds often pop up when we deal with square roots that can’t be simplified to whole numbers – like in measurements or areas involving diagonals. Learning to add and subtract them is a bit like collecting like terms in algebra: once you spot the matching roots, you can tidy them up neatly! [Jump to the questions](#practise-now)

## Practise now

Worksheet preview and key skills 

### Worksheet preview

Practise adding and subtracting surds with this self-marking maths worksheet.

The interactive worksheet below generates questions, gives instant feedback, and lets students record their score.

#### What you’ll practise

- Simplifying surds first where possible.
- Identifying like surds.
- Adding or subtracting coefficients of like surds.
- Keeping unlike surds separate.

Use the interactive worksheet below, or read the [Topic guide](#topic-guide) for the method and worked example.

Enter all answers in their simplest form.

  
## Topic guide

### What this worksheet practises

This worksheet provides practice on adding and subtracting surds. Just like in algebra where you can only add "like terms" (such as 2x + 3x), you can only add or subtract surds if the number inside the square root is exactly the same.

### Key method

To add or subtract surds, you must first simplify them to see if they share a common base surd.

- First, simplify each surd in the expression. Look for the largest square number that divides into the number under the root.
- Second, rewrite the surd as a product of the square number and the remaining factor, and take the square root of the square number outside the radical.
- Finally, collect any "like surds" together by adding or subtracting the numbers outside the square roots, treating the surd itself like a letter in algebra.

### Worked example

**Simplify √12 + √27**

Step 1: Simplify √12\. The largest square number factor is 4.

√12 = √(4 × 3) = √4 × √3 = 2√3.

Step 2: Simplify √27\. The largest square number factor is 9.

√27 = √(9 × 3) = √9 × √3 = 3√3.

Step 3: Add the simplified like surds together.

2√3 + 3√3 = 5√3.

### Common mistakes to avoid

The most common mistake is attempting to add the numbers inside the square roots directly. For example, incorrectly assuming that √2 + √3 equals √5\. You can only ever add the coefficients outside the surds, and only when the numbers inside the square roots are identical.

### Things to remember

The first 10 square numbers are 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100\. Memorising these makes finding the largest square factor of a surd much faster.