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Surds

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Lesson notes

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.

Slides

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Practice questions

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  1. 1.Simplify √32 + √98.

    Medium
    • A√130
    • B11√2
    • C9√2

  2. 2.Rationalise the denominator of 12/√3.

    Easy
    • A4√3
    • B12√3/3
    • C4√3/3
    • D12/3

  3. 3.Simplify √20 + √80, giving your answer in the form a√5.

    Medium
    • A10√5
    • B6√5
    • C2√5
    • D8√5

  4. 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. 5.Rationalise the denominator of 10/√6.

    Medium
    • A(10√6)/6
    • B(5√6)/3
    • C(10√6)/3
    • D5√6

  6. 6.Simplify √6 × √3.

    Easy
    • A3√2
    • B√18
    • C3√3
    • D√9

  7. 7.Simplify fully by rationalising the denominator: 20/√5.

    Medium
    • A4√5
    • B20√5/5
    • C4√5/5
    • D5√5

  8. 8.Simplify fully: √200.

    Easy
    • A10√2
    • B20√10
    • C100√2
    • D10√10

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