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To simplify a surd, find the largest square factor inside the root and take its square root outside. Free IGCSE Maths notes and 6 FAQs.
Surds and Rationalising Denominators is part of Powers, Roots and Standard Form in IGCSE Edexcel Higher Maths.
2 min
Reading time
30
Lessons in this topic
IGCSE Edexcel Higher
Pathway
Reviewed by Revise Right
Content review
To simplify a surd, find the largest square factor inside the root and take its square root outside. Free IGCSE Maths notes and 6 FAQs.
Use this lesson to stay inside Powers, Roots and Standard Form while connecting the detail back to the wider Topic 1: Numbers and the Number System topic.
Read the core explanations from Surds and Rationalising Denominators before testing yourself from memory.
Surds let us keep exact answers instead of rounding too early. They appear naturally in algebra, geometry and trigonometry.
The two skills you need most are simplifying roots using square factors and removing surds from denominators cleanly.
Worked example: Simplifying root 72
Simplify root 72.
Answer: √72 = 6√2
Worked example: Rationalising 3 over root 5
Write 3/root 5 in the form a root b over c.
Answer: 3√5 / 5
When a question asks for an exact answer, leaving a surd in the numerator is usually fine, but leaving one in a simple denominator often is not.
Worked example: Simplify Before Rationalising
Simplify and rationalise 4/root 12.
Answer: 4/root 12 = 2root 3 / 3
Published by Revise Right. Use the official specification reference to check the requirements for your course. Content updated: 2026-03-25.
Move to the related lessons or back to the topic page when you need the wider sequence before exam-style practice.
To simplify a surd, find the largest square factor inside the root, split the root into two roots, then take the square root of the square factor outside. For example, √72 = √36 × √2 = 6√2.
Unlike surds cannot be added or subtracted as if they were ordinary numbers; this is a common pitfall. Every answer should be kept exact, without switching to decimals unless the question asks for an approximation.
√5 × √5 = 5. This is why multiplying the top and bottom of a fraction by √5 removes the surd from a denominator of √5.
First simplify the surd: root 12 = root(4 x 3) = 2 root 3, so 4/root 12 = 2/root 3 once the common factor of 2 cancels. Then multiply top and bottom by root 3 to get 2 root 3 / 3.
To rationalise 3/√5, multiply the whole fraction by √5/√5, which keeps the value unchanged. Because √5 × √5 = 5, the answer is 3√5/5, with no surd in the denominator.
Answers are left in surd form because surds keep the value exact instead of rounding too early. A surd such as √2 or 3√5 is an irrational root written exactly, and a decimal approximation loses that exact value.