National 5 Maths
Surds
Course content
- Simplifying surds
- Rationalising the denominator.
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 \(5.\)
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:
\(\sqrt{3}\), \(4\sqrt{6}\), \(\sqrt[\leftroot{-1}\uproot{3}\scriptstyle 3]{19}\), \(\pi\)
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.
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.
Textbook page references
- TeeJay Maths Book N5 pp.170-172
- Zeta National 5+ Maths pp.5-12
- Leckie National 5 Maths pp.2-12
Self-study course
- National5.com Highly recommended. Save £10 with discount code 'Maths.scot'.
Example 1 (non-calculator)
Simplify \( \sqrt{12} \)
Example 2 (non-calculator)
Simplify \( 3\sqrt{72} \)
Example 3 (non-calculator)
Simplify \( 2\sqrt{20} + 6\sqrt{45} \)
Example 4 (non-calculator)
Simplify \( \sqrt{8} \times \sqrt{12} \)
Find a Maths tutor
Do you need a National 5 Maths tutor?
Click here to find a tutor in your area.
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.
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.
Recommended student books
Zeta Maths: National 5+ practice bookLeckie: National 5 Maths textbook
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 \( \sqrt{10}\left(\sqrt{10}-\sqrt{2}\right)+8\sqrt{5}\)
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 \( \sqrt{75} - \sqrt{3} \)
Need a Nat 5 Maths tutor?
Just use this handy little widget and our partner Bark.com will help you find one.
Maths.scot worksheet
• Surds worksheet • Answer sheet • See all National 5 Maths worksheets |
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) |
Click here to study the surds notes on National5.com.
You may also want to check out their Nat 5 Maths self-study course. We think it's excellent. If you decide to enrol, just use coupon code "Maths.scot" to get £10 discount.
⇦ Nat 5 topic list ⇧ Top of this page
|