N8: Surds
Calculate exactly with fractions, multiples of π; simplify surds, rationalise denominators
📋 Key Concepts
What is a Surd? A surd is an irrational square root that cannot be written as an exact decimal. Examples: √2, √3, √5, √6...
Important: √4 = 2 is NOT a surd (it's rational). √2 ≈ 1.414... is a surd (it's irrational and cannot be written exactly).
Key Surd Rules:
√(a × b) = √a × √b
√(a ÷ b) = √a ÷ √b
(√a)² = a
√a² = a (for a ≥ 0)
√(a × b) = √a × √b
√(a ÷ b) = √a ÷ √b
(√a)² = a
√a² = a (for a ≥ 0)
📝 Simplifying Surds
Method: Find the largest square number that divides into the surd. Split it and simplify.
Example 1
Simplify: √12
Solution:
√12 = √(4 × 3) = √4 × √3 = 2√3
Example 2
Simplify: √50
Solution:
√50 = √(25 × 2) = √25 × √2 = 5√2
Example 3
Simplify: √72
Solution:
√72 = √(36 × 2) = √36 × √2 = 6√2
Alternative: √72 = √(9 × 8) = 3√8 = 3 × 2√2 = 6√2
📝 Adding and Subtracting Surds
Rule: Only "like surds" can be added or subtracted. Simplify first, then combine like terms.
Example 4
Simplify: √12 + √27
Solution:
√12 = 2√3 and √27 = 3√3
2√3 + 3√3 = 5√3
Example 5
Simplify: 3√20 - √45
Solution:
√20 = 2√5, so 3√20 = 6√5
√45 = 3√5
6√5 - 3√5 = 3√5
📝 Multiplying Surds
Rule: Multiply surds by multiplying the numbers inside: √a × √b = √(a × b)
Example 6
Simplify: √2 × √8
Solution:
√2 × √8 = √16 = 4
Example 7
Expand: √3(2 + √12)
Solution:
√3 × 2 = 2√3
√3 × √12 = √36 = 6
Answer: 6 + 2√3
📝 Rationalising Denominators
Why? A fraction should not have a surd in the denominator. Multiply top and bottom to remove it.
Simple Denominators
Method: If the denominator is √a, multiply top and bottom by √a.
Example 8
Rationalise: 1/√2
Solution:
Multiply by √2/√2:
1/√2 × √2/√2 = √2/2
Example 9
Rationalise: 6/√3
Solution:
6/√3 × √3/√3 = 6√3/3 = 2√3
Complex Denominators
Method: If the denominator is a ± √b, multiply by the "conjugate" a ∓ √b.
Example 10
Rationalise: 1/(2 + √3)
Solution:
Multiply by (2 - √3)/(2 - √3):
1(2 - √3) / [(2 + √3)(2 - √3)]
= (2 - √3) / (4 - 3)
= (2 - √3) / 1 = 2 - √3
Example 11
Rationalise: 5/(3 - √2)
Solution:
Multiply by (3 + √2)/(3 + √2):
5(3 + √2) / [(3 - √2)(3 + √2)]
= 5(3 + √2) / (9 - 2)
= 5(3 + √2) / 7 = (15 + 5√2)/7
📝 Exact Values with π
Exact values: Leave answers in terms of π when asked for exact values.
Example 12
Find the exact area of a circle with radius 5 cm
Solution:
A = πr² = π × 5² = 25π cm²
Do NOT calculate 25 × 3.14... - leave as 25π
🧮 Worked Examples — Exam Style
Surds keep answers EXACT. Simplify by finding the largest square factor; multiply under one root; rationalise by multiplying by the conjugate.
Simplify √48.
Largest square factor: 48 = 16 × 3
√48 = √16 × √3 = 4√3
M1 for 16 × 3, A1 for 4√3. Choosing 4 × 12 gives 2√12 — right idea, not finished; simplify again.
Rationalise 5/√3.
Multiply top and bottom by √3: 5√3/(√3 × √3) = 5√3/3
M1 for × √3/√3, A1 for 5√3/3. √3 × √3 = 3 clears the root from the denominator.
Exam wording to expect: "Simplify", "Give your answer in the form a√b", "rationalise the denominator".
❓ Practice Questions
Q1: Simplify: √75
Q2: Simplify: √18 - √8
Q3: Simplify: 2√5 × 3√10
Q4: Rationalise: 3/√5
Q5: Rationalise: 2/(1 + √3)
Q6: Simplify: √50
Q7: Simplify: √27 + √12
✅ Answers
- √75 = √(25 × 3) = 5√3
- √18 = 3√2, √8 = 2√2, so 3√2 - 2√2 = √2
- 2 × 3 × √50 = 6 × 5√2 = 30√2
- 3/√5 × √5/√5 = 3√5/5
- 2(1 - √3)/(1 - 3) = 2(1 - √3)/(-2) = √3 - 1
🧮 Worked Examples — Harder Exam Style
Rationalise and simplify: 6/(3 − √3).
Multiply by (3 + √3)/(3 + √3): numerator 6(3 + √3) = 18 + 6√3
Denominator: 3² − (√3)² = 9 − 3 = 6 (difference of two squares)
(18 + 6√3)/6 = 3 + √3
M1 for the conjugate, M1 for 9 − 3 = 6, A1 for 3 + √3. The difference-of-two-squares denominator is the standard conjugate result — quote it.
📋 Model Answers — Full Working
Every mark comes from the working, not just the answer. These model answers show the lines a mark scheme expects to see.
Q1. 75 = 25 × 3 → 5√3
Q2. 3√2 − 2√2 = √2 (like surds combine like terms)
Q3. 2 × 3 = 6 outside; √5 × √10 = √50 = 5√2 → 6 × 5√2 = 30√2
Q4. 3√5/5
Q5. Multiply by the conjugate (1 − √3): 2(1 − √3)/(1 − 3) = 2(1 − √3)/(−2) = √3 − 1
Q6. 50 = 25 × 2 → 5√2
Q7. 3√3 + 2√3 = 5√3
📌 Key Facts to Memorise
- √(ab) = √a × √b — split off square factors: √72 = 6√2
- Only LIKE surds add: 2√5 + 3√5 = 5√5; √2 + √3 stays put
- √a × √a = a · √a ÷ √b = √(a/b)
- Rationalise single-root: × √b/√b. Bracket denominator: × conjugate (a − √b)/(a − √b)
- Square factors to spot: 4, 9, 16, 25, 36, 49, 64, 100
🗺️ In the Exam
Surds are Higher-tier content sitting in this folder — they appear on Higher Papers 1/2 (non-calculator friendly): 2–3 marks, plus exact trig-value answers (G21).
Links to: exact trig values (G21), Pythagoras (G20), indices (N7).
🧭 Board Focus — OCR
Spec J560. Three papers of 100 marks each: Paper 1 non-calculator, Papers 2 and 3 calculator, at Foundation/Higher tiers. OCR marks problem-solving explicitly, and its papers tend to front-load multi-step worded contexts — practise extracting the mathematics from prose before calculating.
Always check: specification details change — confirm paper structure and set texts against the current specification for your exam year.
🎯 Exam Tips
- Always simplify surds fully - look for the largest square factor
- When adding/subtracting, check all surds are simplified first
- Memorise: (a + √b)(a - √b) = a² - b
- Leave answers as surds unless asked for a decimal
- "Exact value" means leave in terms of π or as a surd
🧠 Problem-Solving Strategies
Problem-Solving
For surd problems, always simplify first before combining. When rationalising, identify whether the denominator is a simple surd (multiply by √a) or a bracketed expression (multiply by the conjugate). For exact value problems, never convert to decimals — leave answers in surd form or in terms of π.
Multi-Step Problem
A rectangle has length 2 + √3 and width 2 − √3. Show that the area is an integer, and find the perimeter in simplified surd form.
Solution: Area = (2 + √3)(2 − √3) = 4 − 3 = 1 (integer ✓). Perimeter = 2(2 + √3) + 2(2 − √3) = 4 + 2√3 + 4 − 2√3 = 8.
⚠️ Common Errors
Watch Out!
1. Wrong: √12 + √3 = √15 (adding inside one square root) Correct: √12 = 2√3, so √12 + √3 = 2√3 + √3 = 3√3
2. Wrong: √(a + b) = √a + √b Correct: √(a + b) ≠ √a + √b — you cannot split a square root over addition
3. Wrong: Rationalising 1/(3 + √2) by multiplying by √2/√2 Correct: Multiply by the conjugate (3 − √2)/(3 − √2) to eliminate the surd from the denominator
✍️ 6-Mark Exam Question
Extended Answer
6 marks: (a) Simplify √28 − √63 + √175. (b) Rationalise the denominator of 4/(3 − √5). (c) Show that (2 + √5)² = 9 + 4√5. Hence find the exact value of (2 + √5)² + (2 − √5)².
(a) √28 = 2√7, √63 = 3√7, √175 = 5√7. So 2√7 − 3√7 + 5√7 = 4√7.
(b) 4/(3 − √5) × (3 + √5)/(3 + √5) = 4(3 + √5)/(9 − 5) = 4(3 + √5)/4 = 3 + √5.
(c) (2 + √5)² = 4 + 4√5 + 5 = 9 + 4√5 ✓. By symmetry, (2 − √5)² = 9 − 4√5. Sum = 9 + 4√5 + 9 − 4√5 = 18.
Mark scheme: 2 marks for (a) with full simplification, 2 marks for (b) with correct conjugate method, 2 marks for (c) including the sum
📊 AO3: Reason & Interpret
Reasoning and Interpretation
A right-angled triangle has legs of length √8 cm and √18 cm.
(a) Show that the hypotenuse has exact length √26 cm.
(b) A student says "The hypotenuse is √8 + √18 = 2√2 + 3√2 = 5√2 cm." Explain why this is wrong.
(c) Compare the exact hypotenuse with the student's answer. Which is longer?
Answers: (a) By Pythagoras: hyp² = √8² + √18² = 8 + 18 = 26, so hyp = √26 cm. (b) The student added the legs, but Pythagoras' theorem requires SQUARING the legs first, then adding, then square rooting. You cannot just add the sides. (c) √26 ≈ 5.10, 5√2 ≈ 7.07. The student's answer is longer, which shows adding sides gives an overestimate of the hypotenuse.
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