Surds
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Notes de leçon
What is a Surd?
- A surd is the square root of a non-square integer, e.g. √5.
- Surds allow exact answers (e.g. 5√2 instead of 7.07...).
Multiplying and Dividing Surds
- Multiplying: √a × √b = √(ab), e.g. √3 × √5 = √15.
- Dividing: √a ÷ √b = √(a/b), e.g. √21 ÷ √7 = √3.
- Factorising: √(ab) = √a × √b, e.g. √35 = √5 × √7.
Adding and Subtracting Surds
- Only add/subtract like surds (same number under root), e.g. 3√5 + 8√5 = 11√5.
- Unlike surds cannot be combined, e.g. 2√3 + 4√6 stays as is.
- Do not add numbers under square roots: √9 + √4 = 3+2=5, not √13.
Simplifying Surds
- Factorise the number using the largest square factor, e.g. √48 = √(16×3) = 4√3.
- Simplify multiple surds separately then collect like terms, e.g. √32 + √8 = 4√2 + 2√2 = 6√2.
- Expand double brackets like algebra, then simplify using (√a)² = a.
Rationalising Simple Denominators
- If denominator is a surd, multiply numerator and denominator by that surd.
- Example: a/√b = (a/√b)×(√b/√b) = a√b/b.
- This removes the surd from the denominator because √b × √b = b.
Rationalising Harder Denominators
- If denominator is a + √b, multiply by its conjugate a – √b.
- Use difference of squares: (a+√b)(a–√b) = a² – b.
- Example: 2/(3+√5) = 2(3–√5)/(9–5) = (6–2√5)/4 = (3–√5)/2.
Key Exam Tips
- If calculator gives a surd, keep it in surd form throughout working.
- After rationalising, check denominator has no surd left.
- Always simplify surds fully before adding/subtracting.
Diapos
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Questions d'entraînement
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1.Simplify √32 + √98.
Medium- A√130
- B11√2
- C9√2
2.Rationalise the denominator of 12/√3.
Easy- A4√3
- B12√3/3
- C4√3/3
- D12/3
3.Simplify √20 + √80, giving your answer in the form a√5.
Medium- A10√5
- B6√5
- C2√5
- D8√5
4.Show that √45 + √20 = 5√5. Which step is correct?
Easy- A√45 = 3√5, √20 = 2√5, sum = 5√5
- B√45 = 9√5, √20 = 4√5, sum = 13√5
- C√45 = 5√5, √20 = 2√5, sum = 7√5
- D√45 = 3√5, √20 = 4√5, sum = 7√5
5.Rationalise the denominator of 10/√6.
Medium- A(10√6)/6
- B(5√6)/3
- C(10√6)/3
- D5√6
6.Simplify √6 × √3.
Easy- A3√2
- B√18
- C3√3
- D√9
7.Simplify fully by rationalising the denominator: 20/√5.
Medium- A4√5
- B20√5/5
- C4√5/5
- D5√5
8.Simplify fully: √200.
Easy- A10√2
- B20√10
- C100√2
- D10√10
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