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GCSE Maths Number Surds

Simplifying surds

Simplifying Surds

Here we will learn about simplifying surds for GCSE maths including the definition of a surd, the rules of surds, and how to write surds in their simplest form.

There are also simplifying surds worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

What is simplifying surds?

Simplifying surds is where we rewrite a surd in its simplest form by ensuring the number underneath the root sign (this number is called the radicand) has no square numbers as factors. We make the number as small as possible by extracting square factors from underneath the root sign.

We can use the three important laws of surds, which can be derived from the laws of indices (powers) to help us to simplify surds:

\[\sqrt{m} \times \sqrt{n}=\sqrt{m n}\]

\[\sqrt{m} \div \sqrt{n}=\sqrt{\frac{m}{n}}\]

\[\sqrt{m} \times \sqrt{m}=m\]

E.g.

\[\begin{aligned} \sqrt{20} &=\sqrt{4 \times 5} \\ &=\sqrt{4} \times \sqrt{5} \\ &=2 \times \sqrt{5} \\ &=2 \sqrt{5} \end{aligned}\]

What is simplifying surds?

How to simplify surds

In order to simplify a surd:

Find a square number that is a factor of the number under the root.
Rewrite the surd as a product of this square number and another number, then evaluate the root of the square number.
Repeat if the number under the root still has square factors.

How to simplify surds

Simplifying surds worksheet

Get your free simplifying surds worksheet of 20+ questions and answers. Includes reasoning and applied questions.

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Simplifying surds worksheet

Get your free simplifying surds worksheet of 20+ questions and answers. Includes reasoning and applied questions.

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Simplifying surds examples

Example 1: A simple integer

Simplify:

\[\sqrt{8}\]

Find a square number that is a factor of the number under the root.

Square numbers are 1, 4, 9, 16, 25, …

4 is a factor of 8 (because 4 × 2 = 8).

2Rewrite the surd as a product of this square number and another number, then evaluate the root of the square number.

\[\begin{aligned} \sqrt{8} &=\sqrt{4 \times 2} \\ &=\sqrt{4} \times \sqrt{2} \\ &=2 \times \sqrt{2} \\ &=2 \sqrt{2} \end{aligned}\]

3Repeat if the number under the root still has square factors.

In this case, there are no square numbers that are factors of 2, so the surd is fully simplified.

\[\sqrt{8}=2 \sqrt{2}\]

Example 2: A simple integer

Simplify:

\[\sqrt{45}\]

Find a square number that is a factor of the number under the root.

Square numbers are 1, 4, 9, 16, 25, …

9 is a factor of 45 (because 9 × 5 = 45).

Rewrite the surd as a product of this square number and another number, then evaluate the root of the square number.

\[\begin{aligned} \sqrt{45} &=\sqrt{9 \times 5} \\ &=\sqrt{9} \times \sqrt{5} \\ &=3 \times \sqrt{5} \\ &=3 \sqrt{5} \end{aligned}\]

Repeat if the number under the root still has square factors.

In this case, there are no square numbers that are factors of 5, so the surd is fully simplified.

\[\sqrt{45}=3 \sqrt{5}\]

Example 3: A larger integer

Simplify:

\[\sqrt{240}\]

Find a square number that is a factor of the number under the root.

Square numbers are 1, 4, 9, 16, 25, …

4 is a factor of 240 (because 4 × 60 = 240).

Rewrite the surd as a product of this square number and another number, then evaluate the root of the square number.

\[\begin{aligned} \sqrt{240} &=\sqrt{4 \times 60} \\ &=\sqrt{4} \times \sqrt{60} \\ &=2 \times \sqrt{60} \\ &=2 \sqrt{60} \end{aligned}\]

Repeat if the number under the root still has square factors.

\[\sqrt{240}=2\sqrt{60}\]

In this case, 4 is a factor of 60 and also a square number, so the surd is not yet fully simplified. We repeat the process, rewriting 60 as the product 4 × 15.

\[\begin{aligned} \sqrt{240} &=2 \sqrt{60} \\ &=2 \sqrt{4 \times 15} \\ &=2 \times \sqrt{4} \times \sqrt{15} \\ &=2 \times 2 \times \sqrt{15} \end{aligned}\]

There are no square numbers that are factors of 15, so we cannot reduce the radicand any further.

To finish off, work out the product of the integers in front of the root (2 × 2 = 4).

The final answer, in its simplest form, is

\[\sqrt{240}=4 \sqrt{15}\]

Note that you could have completed this simplification in one step by spotting that 16 is a square factor of 240, as follows:

\[\begin{aligned} \sqrt{240} &=\sqrt{16 \times 15} \\ &=\sqrt{4} \times \sqrt{15} \\ &=4 \sqrt{15} \end{aligned}\]

This method is clearly more efficient, but it’s not always easy to spot large square factors; in this case, you will probably find it easier to work in stages.

Example 4: A larger integer

Simplify:

\[\sqrt{108}\]

Find a square number that is a factor of the number under the root.

Square numbers are 1, 4, 9, 16, 25, …

9 is a factor of 108 (because 9 × 12 = 108).

Rewrite the surd as a product of this square number and another number, then evaluate the root of the square number.

\[\begin{aligned} \sqrt{108} &=\sqrt{9 \times 12} \\ &=\sqrt{9} \times \sqrt{12} \\ &=3 \times \sqrt{12} \\ &=3 \sqrt{12} \end{aligned}\]

Repeat if the number under the root still has square factors.

\[\sqrt{108}=3 \sqrt{12}\]

In this case, 4 is a factor of 12 and also a square number, so the surd is not yet fully simplified. We repeat the process, rewriting 12 as the product 4 × 3.

\[\begin{aligned} \sqrt{108} &=3 \sqrt{12} \\ &=3 \sqrt{4 \times 3} \\ &=3 \times \sqrt{4} \times \sqrt{3} \\ &=3 \times 2 \times \sqrt{3} \end{aligned}\]

There are no square numbers that are factors of 3, so we cannot reduce the radicand any further.

To finish off, work out the product of the integers in front of the root (3 × 2 = 6).

The final answer in its simplest form is

\[\sqrt{108}=6 \sqrt{3}\]

Example 5: Strategies for very large integers

Simplify:

\[\sqrt{720}\]

Find a square number that is a factor of the number under the root.

16 is a factor of 720 (because 16 × 45 = 720).

Rewrite the surd as a product of this square number and another number, then evaluate the root of the square number.

\[\begin{aligned} \sqrt{720} &=\sqrt{16 \times 45} \\ &=\sqrt{16} \times \sqrt{45} \\ &=4 \times \sqrt{45} \\ &=4 \sqrt{45} \end{aligned}\]

Repeat if the number under the root still has square factors.

\[\sqrt{720}=4 \sqrt{45}\]

In this case, 9 is a factor of 45 and also a square number, so the surd is not yet fully simplified. We repeat the process, rewriting 45 as the product 9 × 5.

\[\begin{aligned} \sqrt{720} &=4 \sqrt{45} \\ &=4 \sqrt{9 \times 5} \\ &=4 \times \sqrt{9} \times \sqrt{5} \\ &=4 \times 3 \times \sqrt{5} \end{aligned}\]

There are no square numbers that are factors of 5, so we cannot reduce the radicand any further.

To finish off, work out the product of the integers in front of the root (4 × 3 = 12).

The final answer in its simplest form is

\[\sqrt{720}=12 \sqrt{5}\]

The prime factor tree method for larger numbers

This is an alternative method for simplifying surds that is very effective for larger numbers. It is similar to, and based on, prime factor trees.

Find any factor pair of the radicand – like a prime factor tree.

10 and 72 are a factor pair with a product of 720.

Continue like a prime factor tree, until the numbers under the square root signs are either primes or squares.

2 and 5 are primes, and 4 and 9 are squares.

(Note that we could also stop at 36 which is also square but going further, to make the numbers smaller, makes them easier to deal with.)

Beginning with the square numbers, write out as a product as you would with a prime factor tree. Then simplify the answer.

Square root both the 4 and the 9 to get 2 and 3 respectively.

\[\sqrt{2} \times \sqrt{2}=\sqrt{4}=2\]

Finally, 2 × 3 × 2 = 12, so the answer is

\[12 \sqrt{5}\]

Example 6: A partially simplified surd expression

Simplify:

\[3 \times 4 \sqrt{54}\]

Remember that this means

\[3 \times 4 \times \sqrt{54}\]

\[12 \times \sqrt{54}\]

So all we need to do is deal with the root 54 as with the previous examples.

Find a square number that is a factor of the number under the root.

9 is a factor of 54 (because 9 × 6 = 54).

Rewrite the surd as a product of this square number and another number, then evaluate the root of the square number.

\[\begin{aligned} 12 \times \sqrt{54} &=12 \times \sqrt{9 \times 6} \\ &=12 \times \sqrt{9} \times \sqrt{6} \\ &=12 \times 3 \times \sqrt{6} \\ &=36 \sqrt{6} \end{aligned}\]

Repeat if the number under the root still has square factors.

In this case, there are no square numbers that are factors of 6, so the surd is fully simplified.

\[3 \times 4 \sqrt{54}=36 \sqrt{6}\]

Common misconceptions

Incorrectly rewriting the number under the square root sign (the radicand) as a product of any two factors

One of these factors must be a square number in order for you to be able to simplify the surd.

Not simplifying fully

Always check that there are no square factors of the number under the root.

Simplifying surds is part of our series of lessons to support revision on surds. You may find it helpful to start with the main surds lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:

Practice simplifying surds questions

\sqrt{3}

3

9

81

The square root of $9$ is $3$ (this is a rational number, so the square root of $9$ is not a surd).

2 \sqrt{10}

4 \sqrt{5}

5 \sqrt{4}

2 \sqrt{5}

$4$ is the largest square factor of $20, (4\times5=20)$ . The square root of $4$ is $2$ .

3 \sqrt{5}

2 \sqrt{15}

\sqrt{15}

5 \sqrt{3}

There are no square numbers that are factors of $15$ , so this surd is already simplified as far as possible.

6\sqrt{5}

\sqrt{150}

10\sqrt{15}

5\sqrt{6}

$25$ is the largest square factor of $150, (25\times6=150)$ . The square root of $25$ is $5$ .

4\sqrt{23}

4\sqrt{92}

2\sqrt{23}

2\sqrt{92}

$16$ is the largest square factor of $368, (16\times23=368)$ . The square root of $16$ is $4$ .

12\sqrt{70}

6\sqrt{70}

6\sqrt{140}

9\sqrt{140}

$4$ is the largest square factor of $280, (4\times70=280)$ . The square root of $4$ is $2$ , so $\sqrt{280}=2 \sqrt{70}$ . Finally, multiply by the $3$ to get $6\sqrt{70}$ .

Simplifying surds GCSE questions

1. Simplify
$\sqrt{84}$

(2 marks)

Show answer

\sqrt{84}=\sqrt{4} \times \sqrt{21}

(1)

2 \sqrt{21}

Correct answer

(1)

2. Simplify fully
$\sqrt{540}$

(2 marks)

Show answer

\sqrt{540}=\sqrt{36} \times \sqrt{15} \text { or } \sqrt{540}=\sqrt{4} \times \sqrt{135}

(1)

$6 \sqrt{15}$
Correct answer

(1)

3. $\sqrt{912}=4\sqrt{k}$
where $k$ is an integer. Find the value of $k$ .

(2 marks)

Show answer

$\sqrt{912}=\sqrt{16}\times \sqrt{k}$
or sight of $912\div16$

(1)

$4 \sqrt{57}$
so $k=57$

(1)

4. Simplify fully
$5 \sqrt{80}$

(2 marks)

Show answer

5 \sqrt{80}=5\times\sqrt{5}\times\sqrt{16}

(1)

$20\sqrt{5}$
Correct answer

(1)

Learning checklist

You have now learned how to:

Identify a surd as an irrational root

Simplify surds

The next lessons are

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Introduction

What is simplifying surds?

How to simplify surds

Simplifying surds worksheet

Simplifying surds examples

↓

1) A simple integer 2) A simple integer 3) A larger integer 4) A larger integer 5) Strategies for very large integers 6) A partially simplified surd expression

Common misconceptions

Related lessons

Practice simplifying surds questions

Simplifying surds GCSE questions

Learning checklist

Next lessons

Still stuck?