Step-by-step worked examples and graded practice questions on surds — simplifying, adding, subtracting and multiplying surds, and rationalising the denominator.
📚 Foundation & Higher✅ 15 Practice Questions🔍 Full Worked Examples⚠️ Common Mistakes
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A surd is a root (usually a square root) that cannot be simplified to a whole number, such as √2 or √7. Because these values are irrational — they never terminate or repeat as a decimal — GCSE questions ask you to leave answers in exact surd form rather than as a rounded decimal.
Two key rules make surds workable:
√a × √b = √(a × b)
√a ÷ √b = √(a ÷ b)
Simplifying surds
Worked Example 1
Simplify √50.
1
Find a square number that divides into 50: 50 = 25 × 2
2
Split the surd using √a × √b = √(a×b): √50 = √25 × √2
3
√25 = 5, so: 5 × √2
Answer5√2
Always look for the largest square factor — this simplifies in one step instead of several.
Adding and subtracting surds
Surds can only be added or subtracted directly if they have the same value under the root sign — just like collecting like terms in algebra.
Worked Example 2
Simplify 3√8 + 2√18
1
Simplify each surd first: √8 = √4 × √2 = 2√2, and √18 = √9 × √2 = 3√2
2
Substitute back: 3(2√2) + 2(3√2) = 6√2 + 6√2
3
Now both terms have √2, so they can be added: 6√2 + 6√2 = 12√2
Answer12√2
Multiplying surds
Worked Example 3
Simplify √3 × √12
1
Multiply under a single root: √3 × √12 = √(3 × 12) = √36
2
36 is a square number: √36 = 6
Answer6
Rationalising the denominator
A fraction is not considered fully simplified if it has a surd on the bottom (denominator). To fix this, multiply the top and bottom by the surd in the denominator — this doesn't change the fraction's value, since you're really multiplying by 1.
Worked Example 4
Rationalise the denominator of 5/√3
1
Multiply top and bottom by √3: (5 × √3) / (√3 × √3)
2
√3 × √3 = 3, so the denominator becomes a whole number: 5√3 / 3
Answer5√3⁄3
Practice questions
Work through each question before checking the answers.
Foundation (Grade 3–5)
Q1Simplify √20.Foundation
Q2Simplify √45.Foundation
Q3Simplify 2√3 + 5√3.Foundation
Q4Simplify √2 × √8.Foundation
Q5Simplify (√5)².Foundation
Higher (Grade 5–7)
Q6Simplify √72.Higher
Q7Simplify 4√2 + 3√8.Higher
Q8Simplify √5 × √20.Higher
Q9Rationalise the denominator of 3/√5.Higher
Q10Expand and simplify (2 + √3)(2 − √3).Higher
Higher — Hard (Grade 8–9)
Q11Expand and simplify (3 + √2)², giving your answer in the form a + b√2.Grade 8–9
Q12Rationalise the denominator of 4/(2 + √3).Grade 8–9
Q13Simplify √48 − √27 + √12.Grade 8–9
Q14Expand and simplify (5 + 2√3)(5 − 2√3).Grade 8–9
Q15Show that 1/(√5 − √3) can be written as (√5 + √3)/2.Grade 8–9
Q15Multiplying top and bottom by (√5 + √3): (√5+√3) ÷ ((√5−√3)(√5+√3)) = (√5+√3) ÷ (5−3) = (√5+√3)/2, as required.
Common mistakes
Common Mistake 1
Not finding the largest square factor
√72 can be split as √4 × √18, but this leaves another surd to simplify. Finding the largest square factor (36) gets straight to the fully simplified answer: 6√2.
Common Mistake 2
Adding surds with different values under the root
√2 + √3 cannot be combined into a single surd — they are not "like terms". Only surds with the same number under the root can be added or subtracted directly.
Common Mistake 3
Forgetting to simplify surds before adding
3√8 + 2√18 doesn't look like it can be simplified until each surd is broken down first — always simplify every surd before checking whether terms can be combined.
Common Mistake 4
Using the wrong multiplier when rationalising a two-term denominator
For a denominator like (2 + √3), multiply by (2 − √3) — the same terms with the opposite sign — not by (2 + √3) again. This creates a "difference of two squares" that eliminates the surd.
Exam tips
💡 Exam Tip 1
List square numbers before simplifying
Keep 4, 9, 16, 25, 36, 49, 64, 81, 100 in mind — quickly scanning for which one divides into your surd's number makes simplifying far faster and less error-prone.
💡 Exam Tip 2
Leave answers as exact surds unless told otherwise
If a question says "give your answer in the form a√b" or "leave your answer as a surd", do not convert to a decimal — a decimal answer will lose marks even if the value is technically correct.
💡 Exam Tip 3
Recognise the difference of two squares pattern
(a + √b)(a − √b) always simplifies to a whole number (a² − b) — spotting this pattern instantly speeds up both rationalising and expanding surd expressions.
💡 Exam Tip 4
Double-check your final answer is fully simplified
After simplifying, check the number under the root has no remaining square factors, and that there's no surd left in the denominator — GCSE mark schemes expect both.
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