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Cambridge IGCSE0580

Surds

Mathematics 0580 Chapter Notes

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Surds
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In this chapter

  1. 1Simplifying a surd
  2. 2Adding and subtracting surds
  3. 3Multiplying and expanding
  4. 4Rationalising a single surd
  5. 5Rationalising a two-term denominator

1. Simplifying a surd

A surd, such as √5, is an exact value. Keep it as a root. Do not replace it with a rounded decimal.

To simplify, find the largest square number that divides the number under the root. Write the number as a product, then take the square root of the square factor. The answer is in simplest form when no square factor is left.

√ab = √a × √b

√20 = √4 × √5 = 2√5

Examiner insight

“Some common errors involved incorrect factorisation and failure to extract perfect square factors.”

Worked example 13 marks

Do not use a calculator. Write √72 in the form a√b, where b is as small as possible.

  1. 1

    largest square factor of 72 is 36: √72 = √(36 × 2) [1]

  2. 2

    = √36 × √2 [1]

  3. 3

    = 6√2 [1]

2. Adding and subtracting surds

Surds add and subtract only when they have the same surd. Simplify each one first, then collect them like algebraic terms. √a + √b is not √(a+b).

a√c + b√c = (a+b)√c

√(200) - √32 = 10√2 - 4√2 = 6√2

Worked example 12 marks

Simplify. √40 + √(160)

  1. 1

    √40 = 2√10 and √(160) = 4√10 [1]

  2. 2

    2√10 + 4√10 = 6√10 [1]

3. Multiplying and expanding

Multiply surds by multiplying the numbers under the roots. To expand brackets, multiply every term by every term and take care with the sign of each product. Then simplify any surds and collect like terms.

√a × √b = √ab

√a × √a = a

(p + √q)(p - √q) = p² - q

Worked example 13 marks

Expand and simplify. (5 + √2)(3 - √8)

  1. 1

    √8 = 2√2, so the second bracket is 3 - 2√2 [1]

  2. 2

    15 - 10√2 + 3√2 - √2 × 2√2, and √2 × 2√2 = 4 [1]

  3. 3

    = 11 - 7√2 [1]

4. Rationalising a single surd

To rationalise the denominator means to remove the surd from the bottom of a fraction. Multiply the numerator and the denominator by the same surd. This multiplies the fraction by 1, so its value does not change.

Simplify the result. Do not try to cancel the surd as it stands in the fraction.

k/(√a) = k/(√a) × (√a)/(√a) = (k√a)/a

10/(√5) = (10√5)/5 = 2√5

Worked example 14 marks

√(29/20) + 6/(√5) = k√5 Find the value of k.

  1. 1

    29/20 = 49/20, so √(49/20) = 7/(2√5) [1]

  2. 2

    7/(2√5) = (7√5)/10 [1]

  3. 3

    6/(√5) = (6√5)/5 = (12√5)/10 [1]

  4. 4

    (19√5)/10, so k = 1.9 [1]

5. Rationalising a two-term denominator

When the denominator is a + √b or a - √b, multiply top and bottom by the same two terms with the sign between them changed. The denominator becomes a whole number, because (a - √b)(a + √b) = a² - b.

Multiply the numerator out as well, then simplify. For example, 1/(√3 - 1) = (1 + √3)/2.

k/(a - √b) = (k(a + √b))/((a - √b)(a + √b)) = (k(a + √b))/(a² - b)

Examiner insight

“The main problem was multiplying the denominators correctly.” “Some candidates made an error with the sign and gave 14; others wrote –16, coming from 9 – 25.”

Worked example 13 marks

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

  1. 1

    multiply top and bottom by 3 + √5: (2(3 + √5))/((3 - √5)(3 + √5)) [1]

  2. 2

    denominator = 9 - 5 = 4 [1]

  3. 3

    (2(3 + √5))/4 = (3 + √5)/2 [1]

Recap

  • Extended: simplify a surd by taking out the largest square factor: √20 = 2√5.
  • Extended: add or subtract only the same surd, after simplifying each one.
  • Extended: √a × √a = a. Expand brackets term by term and watch each sign.
  • Extended: rationalise k/(√a) by multiplying top and bottom by √a.
  • Extended: rationalise k/(a - √b) by multiplying top and bottom by a + √b. The denominator becomes a² - b.

End-of-chapter exercise

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

  1. Do not use a calculator. Write √50 in the form a√2.2 marks
  2. Write √48 in its simplest form.2 marks
  3. Do not use a calculator. Simplify √18 + √8.3 marks
  4. Simplify √75 - √12.3 marks
  5. Do not use a calculator. Expand and simplify (2 + √3)(4 - √3).3 marks
  6. A rectangle measures 2√6 cm by 3√2 cm. Find its exact area. Give your answer as a surd in its simplest form.3 marks
  7. Do not use a calculator. Rationalise the denominator of 6/(√3).2 marks
  8. Rationalise the denominator of 12/(√8) and simplify.3 marks
  9. Do not use a calculator. A strip of wood of length √48 cm is joined end to end to a strip of length 6/(√3) cm. A piece of length 4/(√3 - 1) cm is then cut off. Find the length that remains. Give your answer in the form a√3 + b, where a and b are integers.5 marks
  10. A rectangle has area (5 + √5) cm² and width (3 - √5) cm. Find the perimeter of the rectangle. Give your answer in the form p + q√5, where p and q are integers.6 marks

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