Key Terms
- Surd: An irrational root that cannot simplify to a whole number (e.g. √2, √3)
- Irrational number: Cannot be written as a fraction a/b; infinite non-repeating decimal
- Rationalising the denominator: Removing a surd from the bottom of a fraction
- Conjugate: (a + √b) and (a − √b) — used to rationalise two-term denominators
- Like surds: Surds with the same root (can be added like terms)
Must-Know Facts
- √4 = 2 and √9 = 3 are NOT surds (they simplify to integers)
- √a + √b ≠ √(a+b) — you cannot add different surds
- Only LIKE surds can be added: 3√2 + 5√2 = 8√2
- To simplify: find the largest square factor (e.g. √72 = √(36×2) = 6√2)
- To rationalise 1/√2: multiply top and bottom by √2 → √2/2
- (a+√b)(a−√b) = a² − b (no surd in answer)
Key Surd Rules
- √a × √b = √(ab)
- √a ÷ √b = √(a/b)
- √(a²b) = a√b (take square root of square factor)
- Rationalise: multiply numerator and denominator by the surd in denominator
- Square factors to check: 4, 9, 16, 25, 36, 49, 64, 81, 100
Common Mistakes
- Adding different surds: √2 + √3 ≠ √5 — you can only add like surds (same number under the root sign)
- Missing the largest square factor: √72 = √(4×18) = 2√18 is not fully simplified — spot √(36×2) = 6√2 instead by checking larger square factors
- Multiplying coefficients and surds separately: 3√2 × 2√3 = 6√6 — multiply the integers together and the surds together
- Not rationalising the denominator: Answers with surds in the denominator are not in simplest form — multiply top and bottom by the denominator surd
This topic summary covers Knowledge Organiser: Surds within Surds for GCSE Mathematics. Revise Surds in Number for GCSE Mathematics with 14 exam-style questions and 22 flashcards. This is a high-frequency topic, so it is worth revising until the explanation feels precise and repeatable. It is section 15 of 15 in this topic. Use this topic summary to connect the idea to the wider topic before moving on to questions and flashcards.
Practice questions for Surds
Which of these is the simplified form of √48?
Explain why it is preferable to write fractions in rationalized form rather than leaving a surd in the denominator.
Quick recall flashcards
What is a surd?
An irrational root that cannot be simplified to a whole number
Examples: √2, √3, √5, ∛7
NOT surds: √4 = 2, √9 = 3 (these simplify to whole numbers)
What are Like Surds?
Surds with the same root part
Examples of like surds:
• 3√2 and 5√2 (both have √2)
• 2√7 and -4√7 (both have √7)
Can add/subtract like surds:
3√2 + 5√2 = 8√2