Learning Goal

Syllabus Requirement

Students need to learn the use and manipulation of surds, with the below point being highlighted in particular:

•    Be able to rationalise denominators.

Study Notes

A surd is an unresolved ??n??th root. So, for example, ??\sqrt{5}?? is a surd and ??\sqrt{4}?? is a surd - although the latter can be written so that it is not a surd because ??\sqrt{4} = 2?? and ??2?? is not a surd.

A simple surd is one with just a single term, such as ??\sqrt{3}?? or ??\sqrt{8}??.

A compound surd is the sum of two or more simple surds or a simple surd and a rational number, such as ??\sqrt{3} + 2?? or ??\sqrt{3} + \sqrt{2}??.

The conjugate surd to a compound surd, with two terms, ??\sqrt{a} + \sqrt{b}?? or ??c + \sqrt{d}?? is given by ??\sqrt{a} - \sqrt{b}?? and ??c - \sqrt{d}??, respectively.

There are 4 basic laws that govern surd operations (namely addition, subtraction, multiplication and division) and they are listed in the information box below:

The 4 basic laws of surds.

??a \sqrt{b} + c \sqrt{b} = (a + c) \sqrt{b}??

Law 2: Subtraction of surds
??a \sqrt{b} - c \sqrt{b} = (a - c) \sqrt{b}??

Law 3: Multiplication of surds
??\sqrt{a} \times \sqrt{b} = \sqrt{ab}??

Law 4: Division of surds
?? \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}??

You should also be aware of one common mistake, that students often make:

Common mistake

??\sqrt{a + b} \neq \sqrt{a} + \sqrt{b}??
Make sure that you understand how this statement is different to Law 1.

One method that is often examined and is specifically mentioned on the syllabus is rationalising the denominator. Remember, you should always rationalise the denominator if you want to give the answer in its simplest form.

Rationalising the denominator

Method 1: Simple surd in denominator $$\frac{x}{\sqrt{y}}$$ If there is just a simple surd in the denominator, such as ??\sqrt{y}??, as above, then simply multiply both the numerator and denominator by that surd to remove it. This rationalises the denominator as shown below: $$\frac{x}{\sqrt{y}}$$ $$= \frac{x \times \sqrt{y}}{\sqrt{y} \times \sqrt{y}}$$ $$= \frac{x \sqrt{y}}{y}$$

Method 2: Compound surd in denominator $$\frac{x}{\sqrt{y} + \sqrt{z}}$$ If there is a compound surd with two terms in the denominator, such as ??\sqrt{y} + \sqrt{z}??, then simply multiply both the numerator and denominator by the conjugate surd to remove it. This rationalises the denominator as shown below: $$\frac{x}{\sqrt{y} + \sqrt{z}}$$ $$=\frac{x \times (\sqrt{y} - \sqrt{z})}{(\sqrt{y} + \sqrt{z})(\sqrt{y} - \sqrt{z})}$$ $$=\frac{x \times (\sqrt{y} - \sqrt{z})}{\sqrt{y}\sqrt{y} + \sqrt{y}\sqrt{z} - \sqrt{y}\sqrt{z} - \sqrt{z}\sqrt{z}}$$ $$=\frac{x \times (\sqrt{y} - \sqrt{z})}{y - z}$$

Worked Examples

We'll now run through some examples of each of these rules in turn, with some more advanced examples thereafter.

Law 1

Example 1

Question

Simplify the following expression ?? 3 \sqrt{11} + 2 \sqrt{11} ??.

Law 2

Example 2

Question

Simplify the following expression ?? 9\sqrt{13} - 3 \sqrt{13} ??.

Law 3

Example 3

Question

Simplify the following expression ?? 12\sqrt{108}??.

Law 4

Example 4

Question

Simplify the following expression ?? \sqrt{4} \sqrt{\frac{9}{4}}??.

Multiple Laws

Example 5

Question

Simplify the following expression ?? 12\sqrt{24} - 6 \sqrt{48} ??.

Rationalising the Denominator

Example 6

Question

Rationalise the following expression ?? \frac{3}{2 + \sqrt{5}} ??.

Exam Questions

The table below contains every exam question that has been asked on this topic, this includes normal papers from both January and June sittings, International papers and Specimen papers.

To see an exam question and solution simply click on the load question icon (), the question will appear below the table and the solution can be shown by clicking the "Show Solution" button that also appears.

The full exam paper and mark schemes are also available for download by clicking on the download icons in each row of the table ().

Question
Exam Board Subject Paper Year Month Module Question
No.
Parts Total
Marks
Exam
Paper
Mark
Scheme
Edexcel Maths Standard 2006 January Core 1 5 (a), (b) 6
Edexcel Maths Standard 2006 June Core 1 6 (a), (b) 4
Edexcel Maths Standard 2007 January Core 1 2 (a), (b) 4
Edexcel Maths Standard 2007 June Core 1 1 n/a 2
Edexcel Maths Standard 2008 January Core 1 3 n/a 4
Edexcel Maths Standard 2009 January Core 1 3 n/a 2
Edexcel Maths Standard 2009 June Core 1 1 (a), (b) 4
Edexcel Maths Standard 2010 January Core 1 2 (a), (b) 6
Edexcel Maths Standard 2010 June Core 1 1 n/a 2
Edexcel Maths Standard 2011 January Core 1 3 n/a 4
Edexcel Maths Standard 2012 January Core 1 2 (a), (b) 6
Edexcel Maths Standard 2012 June Core 1 3 n/a 5
Edexcel Maths Standard 2013 January Core 1 3 (a), (b) 6
Edexcel Maths Standard 2013 June Core 1 1 n/a 4
Edexcel Maths International 2013 June Core 1 2 n/a 4

Exam Tips

1. If you're stuck on a question involving surds, there are two things that you can try:
Rationalise any denominators.
Remove any square factors from inside surds (e.g. ??\sqrt{12}?? can be written as ??2\sqrt{3}?? using Law 3).

2. Exam questions are often simple applications of the above laws, for example, most often taking the form of rewriting a given expression in another form, through rationalising the denominator, such as:
Rewrite ??\frac{4}{1+\sqrt{3}}?? in the form ??a + b\sqrt{3}??, where ??a?? and ??b?? are integers.
Rewrite ??\frac{8}{\sqrt{7}+3}?? in the form ??a + b\sqrt{7}??, where ??a?? and ??b?? are integers.

3. Sometimes exam questions involve rewriting a given expression in another form, through the use of Law 3, such as:
Rewrite ??\sqrt{20} + \sqrt{5}?? in the form ??a\sqrt{b}??, where ??a?? and ??b?? are integers.
Rewrite ??\sqrt{32} - \sqrt{8}?? in the form ??a\sqrt{b}??, where ??a?? and ??b?? are integers.

4. Remember, there is no calculator allowed, so the examiners won’t ask you to complete complicated arithmetic. When looking to break apart a surd using Law 3, start by checking whether each square number (e.g. ??4??, ??9??, ??16??, etc.) is a factor, if it is remove it:
??\sqrt{200}?? becomes ??\sqrt{100 \times 2} = \sqrt{100}\sqrt{2} = 10\sqrt{2}??.
??\sqrt{48}?? becomes ??\sqrt{16 \times 3} = \sqrt{16}\sqrt{3} = 4\sqrt{3}??.

5. You should learn and become familiar with the laws above. None of the information from this page is in the materials given to you in the exam ('Mathematical Formulae'). Although surds are sometimes used in other questions, very rarely will a question involve the surd laws on this page and another part of the material.