0580

Surds

Number

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

The word "surd" appears on only 2 of the 43 papers, but "exact" appears on 27 of them — that instruction is the real signal, and a decimal scores nothing where an exact value was asked for. Take out the largest square factor: 2√12 is not fully simplified.

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