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Simplify Surds

In this worksheet, students will simplify surds using their knowledge of square numbers.

'Simplify Surds' worksheet

Key stage:  KS 3

Year:  Year 9 Maths worksheets

Curriculum topic:   Number

Curriculum subtopic:   Understand Integer Powers/Real Roots

Difficulty level:  

Worksheet Overview

A surd is the irrational root of a rational number!


We use surds in calculations as they are more accurate than using rounded values.


A topic where we often see values in surd form would be in geometry.

When finding the length of a missing side using Pythagoras’ Theorem, we complete the calculation by taking the square root of a value. If we write the answer to the nearest decimal or whole number, we aren’t left with an exact value. So, using the root of the value gives us an exact answer – even if it is irrational!


Examples of Surds are √2 , √3, √6 ...


We can simplify surds if a factor of the number under the root sign is a square number.


Example 1

Simplify √27

First, we need to see if  the value is divisible by a square number.

Listing the square numbers helps with a question such as this.

1, 4, 9, 16, 25, 36, 49, 64, 81, 100

Looking at the list, 27 is divisible by 9.


We can now re-write the surd as

√(9 x 3)

= √9 x √3

We can simplify now as the square root of 9 is 3

= 3 x √3


Example 2

Simplify √32

Remember that we are looking for a factor pair that has a square number


= √(16 x 2)

= √16 x √2

= 4√2


Let's have a go at some questions now. If you need a reminder of what to do, just click on the red help button on the screen and it will show you this introduction again.

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