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MasterMaths online tuition: GCSE Maths Surds: Simplifying and Rationalising Denominators

GCSE Maths Surds: Simplifying and Rationalising Denominators

Writer: Ryan Harvey
Ryan Harvey
Oct 12
4 min read

Surds are exact square-root expressions that cannot be simplified to rational numbers, such as √2. GCSE Higher questions use surds to test algebraic structure and precision. Learn to simplify them, combine like terms, multiply expressions and rationalise denominators without turning everything into decimals.



What makes a number a surd?


√9 equals 3, so it is not an irrational surd. By contrast, √2 has an infinite non-repeating decimal expansion, and leaving it as √2 keeps the value exact. You can simplify √50 because 50 = 25 × 2, so √50 = √25 × √2 = 5√2. The essential skill is spotting the largest square factor. Common useful squares are 4, 9, 16, 25, 36, 49 and 64.



Simplify GCSE Maths surds step by step


Consider √72. Write 72 = 36 × 2, then √72 = √36 × √2 = 6√2. For √12, use 12 = 4 × 3, giving 2√3. If you do not immediately spot the largest square factor, breaking the number into smaller factors can still work. The simplification is complete when the remaining number inside the root has no square factor greater than 1.



Adding and subtracting like surds


You can combine 3√2 + 2√2 to get 5√2 because the surd parts match. But 3√2 + 2√3 cannot be combined into 5√5. Simplify each term before combining: √8 + √18 becomes 2√2 + 3√2 = 5√2. This is similar to collecting like algebraic terms. A common exam error is adding numbers inside the roots as if √a + √b equalled √(a+b); in general, it does not.



Multiplying surds and expanding brackets


Using √a × √b = √(ab) for nonnegative a and b, √6 × √3 = √18 = 3√2. For a bracket, √2(3 + √2) = 3√2 + 2. With two brackets, (√3 + 1)(√3 − 1) is a difference of two squares and equals 3 − 1 = 2. Keep ordinary algebra rules in mind: surds follow the same distributive law as other real numbers.



Rationalising a simple denominator


To remove √3 from 5/√3, multiply top and bottom by √3. This gives 5√3/3 because √3 × √3 = 3. The fraction has the same value, but its denominator is rational. With a denominator such as 2 + √3, multiply by the conjugate 2 − √3. The denominator becomes (2 + √3)(2 − √3) = 4 − 3 = 1. Make sure you multiply the numerator as well, otherwise you change the original fraction.



Check common surds exam traps


Avoid replacing exact surds with rounded decimals unless the question explicitly requires an approximation. Do not assume √(a+b) can be split into √a + √b. Check the difference between 2√3 and √(2 × 3), which are not equal. When rationalising, copy the entire denominator into brackets and distribute carefully. A quick numerical estimate can reveal a sign or arithmetic error, but the final answer should retain its exact form if requested.



Frequently asked questions about GCSE Maths surds


Can I use a calculator for surds?

A calculator can help check an estimate, but Higher GCSE questions often require exact simplified expressions.


Are surds just another form of algebra?

They use ordinary algebra rules while retaining square roots in exact form.


Explore connected maths learning topics


The following carefully selected lessons and articles support the wider skills mentioned above. Use them to find the topic that is most relevant to your current difficulty, rather than trying to read every resource at once.



Book a free online maths introduction


If your child needs a clearer explanation or you want to discuss whether one-to-one online maths tuition would be useful, MasterMaths offers an introductory conversation. Share the year group, any recent feedback and one or two questions that have caused difficulty. Ryan can explain how interactive online tuition works and help you decide on an appropriate next step. There is no promise of a fixed grade outcome; the aim is confident, independent understanding.


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