Surds and exact answers
Keep a square root exact until the question asks for a decimal.
Read the lesson: Surds and rationalising
Extract square factors before combining terms: √(a²b) = |a|√b over the reals. Terms combine only when their simplified square-root parts match. Thus 2√3 + 5√3 = 7√3, while √2 + √3 does not become √5.
To rationalise a denominator containing two terms, multiply numerator and denominator by the conjugate. The denominator becomes a difference of squares. Check that the original denominator is nonzero; rationalising does not change that restriction.
Worked example
Simplify 1/(√3 + 1).
- Multiply top and bottom by √3 − 1.
- The denominator is (√3 + 1)(√3 − 1) = 3 − 1 = 2.
Answer: (√3 − 1)/2
Practise surds and exact answers
Course mapping
These specification references show where the topic occurs. The questions cover only some parts of each topic.
- 8365 · core · 1.3: Surds and rationalising denominators
- 7M20 · core · N1.2: Surds and rationalising denominators
- 6993 · core · AL2: Surds and rationalising denominators
- 4PM1 · core · 1C–1D: Surds and rationalising denominators
- 8365 · core · 1.3: Rationalise a conjugate denominator
- 7M20 · core · N1.2: Rationalise a conjugate denominator
- 6993 · core · AL2: Rationalise a conjugate denominator
- 4PM1 · core · 1C–1D: Rationalise a conjugate denominator
- 8365 · core · 1.3: Exact operations on surds
- 7M20 · core · N1.2: Exact operations on surds
- 6993 · core · AL2: Exact operations on surds
- 4PM1 · core · 1C–1D: Exact operations on surds
- 8365 · core · 1.3: Exact rationalisation with denominator restrictions
- 7M20 · core · N1.2: Exact rationalisation with denominator restrictions
- 6993 · core · AL2: Exact rationalisation with denominator restrictions
- 4PM1 · core · 1C–1D: Exact rationalisation with denominator restrictions
Next practice: Fractional and negative indices.