IGCSE Math RevisionExample 1
Simplify \(\sqrt{50} + \sqrt{18}\), giving your answer as a single surd.
- \(\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}\) and \(\sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}\)
- \(5\sqrt{2} + 3\sqrt{2} = 8\sqrt{2}\)
Answer: \(8\sqrt{2}\)
Cambridge IGCSE Maths (0580) · Non-calculator · 1.18
Surds without a calculator
Surds are Extended content and suit a non-calculator paper: the answer is wanted exactly, as a simplified surd, not as a rounded decimal. Expect them inside Pythagoras and trigonometry answers too.
- Extended only
- Paper 2 (Extended)
- 7 questions
- Free, no sign-in
The method
- Know the square numbers up to \(15^2 = 225\): they are the factors you look for.
- Simplify: split the number under the root into a square number times what is left, using the largest square you can: \(\sqrt{72} = \sqrt{36} \times \sqrt{2} = 6\sqrt{2}\).
- Add or subtract only like surds, after simplifying: \(2\sqrt{5} + 3\sqrt{5} = 5\sqrt{5}\).
- Multiply: \(\sqrt{a} \times \sqrt{b} = \sqrt{ab}\) and \(\sqrt{a} \times \sqrt{a} = a\). Expand brackets exactly as in algebra.
- Rationalise \(\dfrac{k}{\sqrt{a}}\) by multiplying top and bottom by \(\sqrt{a}\).
- Rationalise \(\dfrac{k}{b + \sqrt{c}}\) by multiplying top and bottom by \(b - \sqrt{c}\); the bottom becomes \(b^2 - c\).
Worked examples
Example 2
Rationalise the denominator of \(\dfrac{10}{\sqrt{7} - 2}\).
- Multiply top and bottom by \(\sqrt{7} + 2\).
- Bottom: \((\sqrt{7} - 2)(\sqrt{7} + 2) = 7 - 4 = 3\)
- Top: \(10(\sqrt{7} + 2) = 10\sqrt{7} + 20\)
Answer: \(\dfrac{10\sqrt{7} + 20}{3}\)
Common slips
- \(\sqrt{a} + \sqrt{b}\) is not \(\sqrt{a + b}\).
- Using a square factor that is not the largest, then stopping: \(\sqrt{72} = 2\sqrt{18}\) is not fully simplified.
- Multiplying only the bottom of the fraction when rationalising.
- Sign errors in the bottom line: \((3 - \sqrt{5})(3 + \sqrt{5}) = 9 - 5\), not \(9 + 5\).
Practice questions
No calculator. Write down every step, then open the worked answer.
Question 1
Simplify \(\sqrt{200}\).
Show the worked answer
- \(\sqrt{200} = \sqrt{100 \times 2} = 10\sqrt{2}\)
Answer: \(10\sqrt{2}\)
Question 2
Simplify \(\sqrt{27} + \sqrt{48} - \sqrt{3}\).
Show the worked answer
- \(\sqrt{27} = 3\sqrt{3}\) and \(\sqrt{48} = 4\sqrt{3}\)
- \(3\sqrt{3} + 4\sqrt{3} - \sqrt{3} = 6\sqrt{3}\)
Answer: \(6\sqrt{3}\)
Question 3
Expand and simplify \((3 - \sqrt{2})(4 + \sqrt{2})\).
Show the worked answer
- \(12 + 3\sqrt{2} - 4\sqrt{2} - 2\)
- \(= 10 - \sqrt{2}\)
Answer: \(10 - \sqrt{2}\)
Question 4
Write \(\dfrac{15}{\sqrt{5}}\) in the form \(a\sqrt{5}\), where \(a\) is an integer.
Show the worked answer
- Multiply top and bottom by \(\sqrt{5}\): \(\dfrac{15\sqrt{5}}{5}\)
- \(= 3\sqrt{5}\)
Answer: \(3\sqrt{5}\)
Question 5
Expand \((\sqrt{6} + \sqrt{3})^2\). Give your answer in the form \(a + b\sqrt{2}\), where \(a\) and \(b\) are integers.
Show the worked answer
- \((\sqrt{6} + \sqrt{3})^2 = 6 + 2\sqrt{18} + 3\)
- \(\sqrt{18} = 3\sqrt{2}\), so \(2\sqrt{18} = 6\sqrt{2}\).
- \(= 9 + 6\sqrt{2}\)
Answer: \(9 + 6\sqrt{2}\)
Question 6
Write \(\dfrac{1 + \sqrt{3}}{2 - \sqrt{3}}\) in the form \(a + b\sqrt{3}\), where \(a\) and \(b\) are integers.
Show the worked answer
- Multiply top and bottom by \(2 + \sqrt{3}\).
- Bottom: \((2 - \sqrt{3})(2 + \sqrt{3}) = 4 - 3 = 1\)
- Top: \((1 + \sqrt{3})(2 + \sqrt{3}) = 2 + \sqrt{3} + 2\sqrt{3} + 3 = 5 + 3\sqrt{3}\)
Answer: \(5 + 3\sqrt{3}\)
Question 7
A square has an area of \(72\) cm².
Find the perimeter of the square. Give your answer as a surd in its simplest form.
Show the worked answer
- Side \(= \sqrt{72} = \sqrt{36 \times 2} = 6\sqrt{2}\) cm
- Perimeter \(= 4 \times 6\sqrt{2} = 24\sqrt{2}\) cm
Answer: \(24\sqrt{2}\) cm
Every example and question on this page is original, written for IGCSE Math Revision rather than taken from Cambridge papers, and each answer was re-solved independently before publishing. The syllabus is summarised in our own words: check the official 0580 syllabus.
Keep going
- Surds questions for 0580: Core and Extended, each labelled calculator or non-calculator
- Number practice for 0580, then My mistakes and a Skill Builder on what you missed
- Surds notes and the 0580 formula sheet
- 0580 past papers: time yourself on a specimen Paper 2
- All six non-calculator methods
Questions students ask
Which 0580 papers test surds without a calculator?
Surds is Extended content (syllabus 1.18), so Extended candidates can meet it on Paper 2, the Extended non-calculator paper, as well as on Paper 4 with a calculator. Core candidates do not need it.
Are these Cambridge questions?
No. The method is explained in our own words, and every example and question was written for IGCSE Math Revision and checked independently before publishing. For real papers, use the Cambridge copies linked from our 0580 past papers page.