Most tested MM1.2

Manipulating and Simplifying Surds

Surds are expressions involving irrational roots, like √(3), used to maintain exact mathematical values. For the ESAT, you must be proficient in simplifying expressions with surds and, most importantly, in rewriting fractions to remove surds from the denominator.

Part of the ESAT Mathematics 2 syllabus — revision for the Engineering and Science Admissions Test (ESAT), the UAT-UK admissions test for Cambridge, Imperial, Oxford and UCL.

Key points

  • Simplify surds by extracting square number factors. For example, √(50) = √(25 × 2) = 5*√(2).
  • Treat surds like algebraic variables when adding or subtracting: 7*√(3) - 2*√(3) = 5*√(3), but √(3) + √(5) cannot be simplified.
  • When multiplying, remember that √(a) × √(b) = √(a*b) and critically, √(a) × √(a) = a.
  • To rationalise a denominator with two terms, such as (a + b*√(c)), you must multiply both the numerator and denominator by its conjugate, (a - b*√(c)).

Formulae

√(a × b) = √(a) × √(b)

To simplify a single surd by finding its largest square factor. For example, to simplify √(72), find the largest square factor (36) and write it as √(36 × 2) = 6*√(2).

(x + y)(x - y) = x2 - y2

This is the 'difference of two squares' formula. Use it to rationalise a denominator of the form 'a + b*√(c)' by multiplying it by its conjugate 'a - b*√(c)'.

Definitions

Surd
An expression containing an irrational root (usually a square root) that cannot be simplified to a rational number. For example, 5 + √(2) is a surd, but √(64) is not as it simplifies to 8.
Rationalising the Denominator
The process of rewriting a fraction to eliminate any surds from the denominator, without altering the fraction's value. This is achieved by multiplying the numerator and denominator by an appropriate expression.

Worked example

Express (10 + √(12)) / (3 - √(3)) in the form a + b*√(3), where a and b are integers.

  1. 1

    First, simplify any surds in the expression.

    √(12) can be written as √(4 × 3), which simplifies to 2*√(3).

  2. 2

    Substitute this back into the fraction to get (10 + 2*√(3)) / (3 - √(3)).

  3. 3

    To rationalise, multiply the numerator and denominator by the conjugate of the denominator.

    The conjugate of (3 - √(3)) is (3 + √(3)).

  4. 4

    Numerator expansion:

    (10 + 2*√(3)) × (3 + √(3)) = 10*3 + 10*√(3) + 2*√(3)*3 + 2*√(3)*√(3) = 30 + 10*√(3) + 6*√(3) + 2*3.

  5. 5

    Simplify the numerator by collecting like terms:

    (30 + 6) + (10*√(3) + 6*√(3)) = 36 + 16*√(3).

  6. 6

    Denominator expansion (using difference of two squares):

    (3 - √(3)) × (3 + √(3)) = 32 - (√(3))2 = 9 - 3 = 6.

  7. 7

    Combine the simplified numerator and denominator:

    (36 + 16*√(3)) / 6.

  8. 8

    Divide each term in the numerator by 6:

    36/6 + (16*√(3))/6 = 6 + (8/3)*√(3).

    Let's re-run that with friendlier numbers.

    Prompt should be (10 + 2*√(3)) / (4 - √(3)).

    Ah, wait, let's correct the example.

    Let's make the prompt (15 + 7*√(3)) / (4 + √(3)).

  9. 9

    CORRECTION:

    New prompt:

    Express (15 + 7*√(3)) / (4 + √(3)) in the form a + b*√(3).

  10. 10

    Multiply numerator and denominator by the conjugate of (4 + √(3)), which is (4 - √(3)).

  11. 11

    Numerator:

    (15 + 7*√(3))(4 - √(3)) = 60 - 15*√(3) + 28*√(3) - 7*3 = (60 - 21) + (-15 + 28)*√(3) = 39 + 13*√(3).

  12. 12

    Denominator:

    (4 + √(3))(4 - √(3)) = 42 - (√(3))2 = 16 - 3 = 13.

  13. 13

    Result:

    (39 + 13*√(3)) / 13 = 39/13 + 13*√(3)/13 = 3 + √(3).

Answer: 3 + √(3)

Common mistakes

  • ×Sign errors are very common when expanding brackets, especially when multiplying a positive term by a negative conjugate. For example, in (5 + √(2))(3 - √(2)), ensure the cross-terms are correctly signed: -5*√(2) and +3*√(2).
  • ×A frequent calculation error is squaring a term like (2*√(5))2 and getting 10. The correct calculation is (22) × (√(5))2 = 4 × 5 = 20.
  • ×Forgetting to multiply the entire numerator by the conjugate, not just one term. The whole expression, e.g., (10 + √(2)), must be placed in brackets.

No-calculator tips

  • ✓To simplify large surds like √(288), quickly test for the biggest, most obvious square factors. 288 is even, so test 4: 288 = 4 × 72. 72 is 36 × 2. So √(288) = √(4 × 36 × 2) = 2 × 6 × √(2) = 12*√(2).
  • ✓When rationalising, deal with the integer part and the surd part of the numerator expansion separately to keep your working clear and avoid arithmetic slips.
  • ✓Do a quick sanity check. If you rationalise 10 / (3 + √(2)), you know √(2) is ~1.4, so the denominator is ~4.4. The answer should be a bit more than 2. If you get a large number or a negative one, you've made a mistake.

Read this topic in the official UAT-UK ESAT guide →

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