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# Multiplying surds
- URL: https://www.esheets.io/multiplying-surds/
- Published: 2025-03-20T19:08:07.000Z
- Updated: 2026-06-21T16:46:57.000Z
- Author: Richard Linnington
- Tags: Maths, Number

Multiplying surds comes up when we deal with roots in areas like geometry and physics, especially when working with measurements that aren’t whole numbers. It’s a key skill for simplifying messy square roots into neater expressions – perfect for when you're tackling problems involving areas, distances, or even Pythagoras' Theorem! [Jump to the questions](#practise-now)

## Practise now

Worksheet preview and key skills 

### Worksheet preview

Practise multiplying 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

- Multiplying numbers under square root signs.
- Simplifying the resulting surd where possible.
- Using square factors to simplify.
- Keeping exact surd form.

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

## Topic guide

### What this worksheet practises

This worksheet provides practice on multiplying surds (square roots that do not result in a whole number). Surds behave very similarly to algebraic terms like 'x' or 'y', but they have one special multiplication rule that makes them unique.

### Key method

The fundamental rule of multiplying surds is: **√a × √b = √(a × b)**. You can combine them under a single root.

- Multiply the numbers that are *outside* the square roots together.
- Multiply the numbers that are *inside* the square roots together, keeping the result inside a new square root.
- Combine these two parts into your answer.
- **Simplify (Important):** Look at the number inside your new square root. Can it be divided by a square number (4, 9, 16, 25...)? If so, you must simplify the surd.

### Worked example

**Calculate 3√2 × 4√5.**

Step 1: Multiply the outside numbers.

3 × 4 = 12.

Step 2: Multiply the inside numbers.

√2 × √5 = √10.

Step 3: Combine them.

The final answer is 12√10.

### Worked example 2 (Simplifying)

**Calculate √6 × √8.**

Step 1: Multiply the insides.

√6 × √8 = √48.

Step 2: Simplify √48\. The largest square number that goes into 48 is 16.

√48 = √(16 × 3) = √16 × √3

Because √16 is exactly 4, the final simplified answer is 4√3.

### Common mistakes to avoid

A common error is trying to add the numbers inside the surd instead of multiplying them (writing √6 × √8 = √14). Another frequent mistake occurs when a surd is multiplied by itself. Remember that √5 × √5 = √25, which is exactly 5\. Multiplying a square root by itself just "pops" the number out of the root.

### Things to remember

While you **can** multiply surds together under one roof (√2 × √3 = √6), you **cannot** do this for addition. √2 + √3 does **not** equal √5\. Addition requires the surds to be completely identical, like collecting like terms in algebra.