National 5 Maths
Surds

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Course content

  • Simplifying surds
  • Rationalising the denominator.
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Surd

A surd is a number that cannot be simplified to remove a root sign.

Surds are examples of irrational numbers. An irrational number cannot be written as a fraction with integer numerator and denominator.

At Nat 5, we are mainly interested in square roots, although surds also include cube roots, fourth roots, etc.

Examples of surds: \(\sqrt{3}\), \(4\sqrt{6}, \frac{3\sqrt{5}}{2} \)

\(\sqrt{25}\) is not a surd because 25 is a square number so \(\sqrt{25}\) can simplify to \(5.\)

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Integer

The set of integers is the set of whole numbers {0,1,2,3,...} together with their negative counterparts {-1,-2,-3,...}.

Note that every integer is also rational because it may easily be written as a fraction with numerator 1. For example, \(5 = \frac{5}{1}.\)

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Rational

A number is rational if it can be written as a fraction with integer denominator and numerator.

Examples of rational numbers:
\(3\) (it's \(\frac{3}{1}\)), \(\frac{4}{5}\), \(-2\frac{1}{2}\), 0.9 (it's \(\frac{9}{10}\))

Numbers which cannot be written like this are called irrational.

Examples of irrational numbers:
\(\sqrt{3}\), \(4\sqrt{6}\), \(\sqrt[\leftroot{-1}\uproot{3}\scriptstyle 3]{19}\), \(\pi\)

Textbook page references

Key ideas

  • Surds are just numbers that involve root signs where there is no way to write them without the root sign.
    • \(\sqrt 3\) and \(\sqrt {10}\) are surds.
    • \(\sqrt 4\) and \(\sqrt 9\) are not surds.
  • In a similar way that fractions have a simplest form, surds can also be simplified.
  • "Rationalising the denominator" is a process to make sure that there is no surd on the bottom of a fraction.

Square numbers

  • This topic needs strong knowledge of your times tables.
  • You should also memorise these square numbers : 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144.
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Square number

Squaring a number just means multiplying it by itself.

A square number is obtained by squaring an integer

Examples: \(0^2=0\), \(1^2=1\), \(2^2=4\), \(3^2=9\), \(4^2=16\), etc.

Square numbers are very useful when simplifying surds because the square root of a square number is an integer.

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Example 1 (non-calculator)

Simplify \( \sqrt{12} \)

Example 2 (non-calculator)

Simplify \( 3\sqrt{72} \)

Example 3 (non-calculator)

Simplify \( 2\sqrt{20} + 6\sqrt{45} \)

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Example 4 (non-calculator)

Simplify \( \sqrt{8} \times \sqrt{12} \)

Example 5 (non-calculator)

Rationalise the denominator and simplify: \(\large\frac{4\sqrt{20}}{\sqrt8}\normalsize \)

Example 6 (non-calculator)

SQA National 5 Maths 2014 P1 Q8

Express \( \sqrt{40}+4\sqrt{10}+\sqrt{90} \) as a surd in its simplest form.

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Example 7 (non-calculator)

SQA National 5 Maths 2018 P1 Q11

Express \(\large\frac{9}{\sqrt{6}}\) with a rational denominator. Give your answer in its simplest form.

Example 8 (non-calculator)

SQA National 5 Maths 2019 P1 Q12

Express \(\large\frac{\sqrt{2}}{\sqrt{40}}\) as a fraction with a rational denominator. Give your answer in its simplest form.

Example 9 (non-calculator)

SQA National 5 Maths 2022 P1 Q13

Expand and simplify \( \sqrt{10}\left(\sqrt{10}-\sqrt{2}\right)+8\sqrt{5}\)

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How to Pass National 5 Maths 
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Example 10 (non-calculator)

SQA National 5 Maths 2023 P1 Q8

Express \(\large\frac{12}{\sqrt{15}}\) with a rational denominator. Give your answer in its simplest form.

Example 11 (non-calculator)

SQA National 5 Maths 2024 P1 Q6

Simplify \( \sqrt{75} - \sqrt{3} \)

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Maths.scot worksheet

Surds worksheet
Answer sheet
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Past paper questions

All past paper questions by topic
Simplifying surds:
2014 Paper 1 Q8
2017 Spec. P1 Q3 (with vectors)
2017 Paper 1 Q12 (with statistics)
2018 Paper 1 Q19 (with quadratics)
2021 Paper 1 Q9
2022 Paper 1 Q13
2024 Paper 1 Q6
Rationalising the denominator:
2013 Specimen Paper 1 Q5
2015 Paper 1 Q13
2016 Paper 1 Q9 (with functions)
2018 Paper 1 Q11
2019 Paper 1 Q12
2023 Paper 1 Q8

Other great resources

Videos - Maths180.com
Videos - Larbert High School
1. What are surds?
2. Adding and subtracting
3. Multiplying surds
4. Simplifying surds
5. Dividing
6. Rationalising the denominator
Videos - Mr Graham Maths
1. Simplifying surds
2. Adding and subtracting
3. Rationalising the denominator
Videos - YouKenMaths
1. Simplifying surds
2. Multiplying and dividing
3. Exam questions
PowerPoint - MathsRevision.com
Revision notes - BBC Bitesize
Test yourself - BBC Bitesize
Notes - D R Turnbull
Notes - Maths4Scotland
Notes - National5.com
Lesson notes - Maths 777
1. Simplifying surds 1
2. Simplifying surds 2
3. Simplifying surds 3
4. Rationalising the denominator 1
5. Rationalising the denominator 2
Worked examples - Maths Mutt
Practice questions - Maths Hunter
Essentials Skills worksheets
Simplifying surds (Answers)
Rationalising denominator (Answers)
Worksheet - St Andrew's Academy
Exercise on page 27
Worksheet - Airdrie Academy
Exercises - Larkhall Academy
Pages 2-8 Ex 1-5 (no answers)
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