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Cambridge IGCSE0607

Surds

International Mathematics 0607 Chapter Notes

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Surds
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In this chapter

  1. 1Surds and the square-root rules
  2. 2Adding and subtracting surds
  3. 3Multiplying surds and expanding brackets
  4. 4Rationalising a single surd
  5. 5Rationalising a two-term denominator
  6. 6Simplifying after rationalising

1. Surds and the square-root rules

A surd is a square root that does not work out exactly, such as √5 or √20. In the exam, leave it as a surd and do not round it.

Two rules let you simplify a surd. The root of a product is the product of the roots. A square factor comes out of the root as its square root.

The number outside the root is a coefficient: 2√5 means 2 × √5.

√a × √b = √ab

√(a²b) = a√b

Examiner insight

“There was little understanding shown of simplifying surds in general.”

Worked example 12 marks

Do not use a calculator. Write √48 in the form a√b, where a and b are integers and b is as small as possible.

  1. 1

    find the largest square factor: √48 = √(16 × 3) [1]

  2. 2

    = √16 × √3 = 4√3 [1]

2. Adding and subtracting surds

Only like surds can be added or subtracted: surds with the same number under the root. 5√3 + 4√3 = 9√3, but √2 + √3 cannot be combined.

Simplify every surd first, then collect. Work from left to right. Give the final answer as one simplified surd.

The syllabus example is √(200) - √32 = 10√2 - 4√2 = 6√2.

p√n + q√n = (p+q)√n

Examiner insight

“However, a significant number of candidates correctly found the three surds but were unable to simplify to the final answer, with 0 being a popular incorrect answer.”

Worked example 13 marks

Simplify. √75 + √48 - √12

  1. 1

    write each with a square factor: √75 = 5√3, √48 = 4√3, √12 = 2√3 [1]

  2. 2

    collect like surds: 5√3 + 4√3 - 2√3 [1]

  3. 3

    = 7√3 [1]

3. Multiplying surds and expanding brackets

Multiply surds with √a × √b = √ab. A surd times itself is the number under the root.

To expand brackets, multiply every term by every term. Write each step. When the brackets are (a + √b)(a - √b), the middle terms cancel and the surd disappears.

(√a)² = a

(a + √b)(a - √b) = a² - b

Examiner insight

“The use of brackets should be encouraged in order to multiply out surds correctly. Candidates often tried to do too many steps at once resulting in mistakes being made.”

Worked example 13 marks

Do not use a calculator. Expand and simplify (3 + √5)(2 + √5).

  1. 1

    multiply each term by each term: 6 + 3√5 + 2√5 + √5√5 [1]

  2. 2

    √5 × √5 = 5 [1]

  3. 3

    = 11 + 5√5 [1]

4. Rationalising a single surd

To rationalise the denominator is to rewrite a fraction so that the denominator has no surd in it.

Multiply the numerator and the denominator by the surd in the denominator. This is multiplying by 1, so the value does not change. Then cancel.

The syllabus example is 10/(√5) = (10√5)/5 = 2√5.

p/(√n) = p/(√n) × (√n)/(√n) = (p√n)/n

Worked example 12 marks

Do not use a calculator. Rationalise the denominator of 14/(√7).

  1. 1

    multiply top and bottom by √7: (14√7)/7 [1]

  2. 2

    cancel: 2√7 [1]

5. Rationalising a two-term denominator

When the denominator is a + √b, multiply top and bottom by a - √b. Change the sign between the terms. This is the conjugate.

The denominator becomes a² - b, with no surd. Multiply the numerator term by term. Then cancel any common factor.

The syllabus example is 1/(-1 + √3) = (1 + √3)/2.

p/(a + √b) = (p(a - √b))/(a² - b)

Examiner insight

“Candidates understood the method to rationalise the fraction. The only slips occurred in the simplification or cancelling by a factor of 2.” “Many started part (b) correctly but had problems multiplying surds.”

Worked example 13 marks

Rationalise the denominator and simplify. 5/(4 - 2√3)

  1. 1

    multiply top and bottom by 4 + 2√3 [1]

  2. 2

    denominator: 4² - (2√3)² = 16 - 12 = 4. Numerator: 5(4 + 2√3) = 20 + 10√3 [1]

  3. 3

    cancel the factor 2: (20 + 10√3)/4 = (10 + 5√3)/2 [1]

6. Simplifying after rationalising

A rationalised answer is not finished until it is fully simplified. Check whether one number divides every term of the numerator and the denominator.

When the numerator has a bracket, cancel the common factor with the whole bracket. Then you may leave the answer as one fraction or split it into terms.

Examiner insight

“The common error made by candidates in part (b) was not cancelling throughout by 7 and hence giving an unsimplified answer which scored 2 marks out of 3.”

Worked example 13 marks

Rationalise the denominator and simplify. 21/(4 - √2)

  1. 1

    multiply top and bottom by 4 + √2 [1]

  2. 2

    denominator 16 - 2 = 14, so (21(4 + √2))/14 [1]

  3. 3

    cancel by 7: (3(4 + √2))/2 = (12 + 3√2)/2 [1]

Recap

  • A surd is left as a root. Do not round it when an exact answer is needed.
  • √a√b = √ab and √(a²b) = a√b. Take out the largest square factor.
  • Simplify every surd first. Then add or subtract only like surds.
  • (√a)² = a and (a + √b)(a - √b) = a² - b.
  • Rationalise p/(√n) by multiplying top and bottom by √n.
  • Rationalise p/(a + √b) with the conjugate a - √b. Then cancel.

End-of-chapter exercise

Test yourself on the whole chapter. Work through these before moving on.

  1. Write √45 in the form a√b, where b is as small as possible.2 marks
  2. Do not use a calculator. Simplify √98 - √18.2 marks
  3. Do not use a calculator. Simplify √12 + √27 - √3.3 marks
  4. Do not use a calculator. Work out (2√3)².2 marks
  5. Write √6 × √15 in the form a√b, where b is as small as possible.3 marks
  6. Do not use a calculator. Expand and simplify (5 + √2)(5 - √2).2 marks
  7. Do not use a calculator. Rationalise the denominator of 12/(√6). Give the answer in its simplest form.2 marks
  8. Do not use a calculator. Rationalise the denominator and simplify 6/(3 + √3).3 marks
  9. A rectangular tile has length (√50 + √3) cm and width (√18 - √3) cm. Find the exact perimeter in the form a√2. Then find the exact area in the form p + q√6.5 marks
  10. A triangle has area (3 + √3) cm² and base (4 + 2√3) cm. Find the exact perpendicular height. Give the answer in the form a + b√3.5 marks

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