Most tested M2.11

Surds and Exact Calculations

This topic covers exact calculations involving fractions, surds (irrational roots), and multiples of π. Mastery is essential for problems requiring precise answers without a calculator, a core skill in engineering and science.

Part of the ESAT Mathematics 1 syllabus — revision for the Engineering and Science Admissions Test (ESAT), the UAT-UK admissions test for Cambridge, Imperial, Oxford and UCL.

Key points

  • To add or subtract fractions, first find a common denominator, ideally the lowest common multiple.
  • Simplify surds by extracting the largest possible square number factor. For example, √48 = √(16 × 3) = 4√3.
  • You can multiply and divide surds (√a × √b = √(ab)), but you cannot simply add or subtract them (√a + √b ≠ √(a+b)).
  • To 'rationalise the denominator' means to remove any surds from the bottom of a fraction.
  • For denominators with two terms, like (a + √b), multiply the numerator and denominator by its 'conjugate', (a - √b), to create a difference of two squares.
  • When a question requires an 'exact' answer involving circles or trigonometry, leave π in your final answer rather than using a decimal approximation.

Formulae

√(x) × √(y) = √(x*y)

To combine multiple surds under a single square root when multiplying.

(a + √(b)) × (a - √(b)) = a2 - b

To quickly calculate the new denominator when rationalising. This is the 'difference of two squares' principle.

Definitions

Surd
An expression containing an irrational root, such as √2 or 1+√5, which cannot be simplified into a whole number or fraction.
Rationalise the denominator
The process of rewriting a fraction so that the denominator is a rational number (i.e., it contains no surds).
Conjugate
For a two-term expression involving a surd, such as 3 - √7, its conjugate is 3 + √7. Multiplying an expression by its conjugate eliminates the surd.

Worked example

Express the fraction (√75 - 1) / (2 + √3) in the form a + b√3, where a and b are integers.

  1. 1

    First, simplify the surd in the numerator:

    √75 = √(25 × 3) = 5√3

    The expression becomes (5√3 - 1) / (2 + √3).

  2. 2

    To rationalise, multiply the numerator and denominator by the conjugate of the denominator, which is (2 - √3).

  3. 3

    Numerator expansion:

    (5√3 - 1)(2 - √3) = (5√3 × 2) - (5√3 × √3) - (1 × 2) + (1 × √3) = 10√3 - 15 - 2 + √3.

  4. 4

    Collect terms in the numerator:

    (10√3 + √3) + (-15 - 2) = 11√3 - 17.

  5. 5

    Denominator calculation using difference of squares:

    (2 + √3)(2 - √3) = 22 - (√3)2 = 4 - 3 = 1.

  6. 6

    The final expression is (11√3 - 17) / 1, which simplifies to -17 + 11√3.

Answer: -17 + 11√3

Common mistakes

  • ×Sign errors are frequent when expanding brackets during rationalisation. Forgetting that a negative times a negative is a positive is a common mistake.
  • ×Incorrectly squaring a term with a coefficient. For example, when calculating (2√5)2, a common error is to get 10 instead of the correct 22 × (√5)2 = 4 × 5 = 20.
  • ×Making basic arithmetic errors when dealing with multiple terms, especially when collecting like terms in the numerator after expansion.
  • ×Trying to 'cancel' terms inappropriately, for example, cancelling the √3 in (3 + √3) / √3, which is incorrect.

No-calculator tips

  • ✓To simplify a surd like √N, mentally test for divisibility by small square numbers (4, 9, 25, etc.) to find the largest square factor.
  • ✓When expanding brackets with surds, use a systematic method like FOIL (First, Outer, Inner, Last) to ensure no terms or signs are missed.
  • ✓For fraction arithmetic with awkward numbers, find the Lowest Common Multiple (LCM) for the denominator to keep the numbers as small and manageable as possible.

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

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