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Mathematics 4024

Quick revision notes · Mathematics 4024 · Chapter 32 of 35

Surds

Exact roots: simplifying, calculating with surds, and rationalising denominators.

3 sectionsWorked examples Common mistakes
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At a glance

Exact roots: simplifying, calculating with surds, and rationalising denominators.

  1. Simplifying surds
  2. Adding, multiplying and expanding
  3. Rationalising the denominator

1Simplifying surds

  • A surd is a root that is irrational, like √2 or √12. It is exact — keep it as a surd unless told to round.
  • Find the largest square factor: √72 = √(36 × 2) = 6√2.
  • Square factors to look for: 4, 9, 16, 25, 36, 49, 64, 81, 100.

Formulas

√(ab) = √a × √b
√(a/b) = √a / √b
√a × √a = a

Worked example

Simplify √75 + √12.

  1. √75 = √25 × √3 = 5√3.
  2. √12 = √4 × √3 = 2√3.
  3. 5√3 + 2√3.

7√3

2Adding, multiplying and expanding

  • Only like surds add or subtract: 6√2 − 4√2 = 2√2, but √2 + √3 cannot be combined.
  • Multiply: √3 × √12 = √36 = 6; 2√5 × 3√2 = 6√10.
  • Expand brackets just as in algebra, then simplify each term.
  • (a + √b)(a − √b) = a2 − b: the surds cancel. This is used to rationalise.

Worked example

Expand and simplify (3 + √2)2.

  1. (3 + √2)(3 + √2) = 9 + 3√2 + 3√2 + 2.
  2. Collect: 11 + 6√2.

11 + 6√2

Exam tip: (a + √b)2 is NOT a2 + b — there is a middle term 2a√b.

3Rationalising the denominator

  • 'Rationalise' means: no surd on the bottom of the fraction.
  • a/√b: multiply top and bottom by √b. 10/√5 = 10√5/5 = 2√5.
  • a/(b + √c): multiply top and bottom by (b − √c). a/(b − √c): multiply by (b + √c).

How to do it

  1. Look at the bottom.
  2. Single surd √b → multiply top and bottom by √b.
  3. b ± √c → multiply top and bottom by the same with the opposite sign.
  4. Expand and simplify; cancel any common factor.

Worked example

Rationalise 6/(3 − √3).

  1. Multiply by (3 + √3)/(3 + √3).
  2. Bottom: 9 − 3 = 6.
  3. Top: 6(3 + √3).
  4. 6(3 + √3)/6.

3 + √3

✗Common mistakes

Don’t writeWrite instead
√9 + √16 = √25 √9 + √16 = 3 + 4 = 7
Leaving √72 as 2√18 Use the largest square factor: 6√2
(3 + √2)2 = 9 + 2 Middle term: 9 + 6√2 + 2 = 11 + 6√2
Multiplying only the top when rationalising Multiply top and bottom by the same surd

★In the exam

  • 'Give your answer in the form a + b√c' — make sure a, b, c are integers and the surd is simplified.
  • Surd questions are on the non-calculator paper; check by squaring your answer mentally.
  • Exact answers (like in Pythagoras) are often wanted as simplified surds.