MathematicsExam code: 0580

Surds

Number

PractiseFlashcards

Why it is done

  • A fraction with a root on the bottom has an irrational denominator
  • Rationalising the denominator rewrites it as an equivalent fraction whose denominator is rational
  • The numerator may still contain a root; only the denominator has to be cleared

A single root on the bottom

  • Multiply numerator and denominator by that root
  • This multiplies the fraction by 1, so its value is unchanged, and the denominator becomes a whole number
    • 6/√3 becomes 6√3/3, which simplifies to 2√3

A sum or difference on the bottom

  • Multiply numerator and denominator by the same expression with the middle sign reversed
  • The denominator then expands as a difference of two squares, and both roots vanish
    • (a + √b)(a − √b) = a² − b

Worked example

Rationalising a two-term denominator

Rationalise the denominator of 5 ÷ (4 + √2), giving the answer in its simplest form.

Solution:

  • Reverse the sign in the denominator and multiply top and bottom by 4 − √2
  • The numerator becomes 5(4 − √2) = 20 − 5√2
  • The denominator is a difference of two squares: 4² − (√2)² = 16 − 2 = 14
  • The result is (20 − 5√2) / 14
  • The denominator now contains no root, which is the check that the work is finished

Exam tip

Finish every surd answer in its simplest form

After rationalising, cancel any common factor: (24 + 6√10)/6 must finish as 4 + √10. Stopping while a factor still cancels scored one mark less. The same last mark is lost in other topics on the non-calculator paper when a length, volume or quadratic root asked for in its simplest surd form, or in a given form such as a√b, keeps a square factor under the root or a root on the bottom: √72 and 12/√2 must both become 6√2.

Worked example

Rationalising two denominators

Rationalise the denominator of 6/√3, and of 5/(2 + √3).

Solution:

  • For a single root, multiply top and bottom by that root
  • 6/√3 × √3/√3 = 6√3/3
  • Cancel: 2√3
  • For a two-term denominator, multiply by the same expression with the middle sign changed
  • 5/(2 + √3) × (2 − √3)/(2 − √3)
  • The denominator becomes (2 + √3)(2 − √3) = 4 − 2√3 + 2√3 − 3 = 4 − 3 = 1
  • The two middle terms always cancel, which is the whole point of changing the sign
  • The numerator is 5(2 − √3) = 10 − 5√3
  • The answer is 10 − 5√3, over a denominator of 1

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