Sheet № 80 · Higher only · AQA · Edexcel · OCR
Rationalising the Denominator –
Rationalising the denominator is a Higher tier skill that appears regularly on GCSE Maths exams. Examiners expect answers to have no surds in the denominator, so knowing how to remove them is essential for earning full marks on surd questions.
§Key definitions
Question:
Rationalise the denominator of 6/√3. Give your answer in its simplest form.
Q1 (Higher):
Rationalise the denominator of 10/√5. Simplify your answer.
Q2 (Higher):
Rationalise the denominator of 4/(1 + √3). Give your answer in the form a + b√3.
Q3 (Higher):
Rationalise the denominator of (3 + √2)/(√2). Simplify fully.
§Formulas to memorise
To rationalise a/√b, multiply top and bottom by √b: a/√b = a√b / b
To rationalise 1/(a + √b), multiply by (a − √b)/(a − √b), giving (a − √b) / (a² − b)
(a + √b)(a − √b) = a² − b — difference of two squares removes the surd
Single surd denominator: — Multiply the numerator and denominator by the surd in the denominator.
Two-term denominator (e.g. a + √b): — Multiply numerator and denominator by the conjugate (a − √b).
Worked example
Rationalise the denominator of 6/√3. Give your answer in its simplest form.
Working:
⚠ Common mistakes
- ✗Multiplying only the numerator by the surd. You must multiply both the numerator and the denominator to keep the fraction equivalent. Multiplying top and bottom by the same thing is equivalent to multiplying by 1.
- ✗Using the wrong conjugate. The conjugate of (a + √b) is (a − √b), not (a + √b). Using the same expression will not eliminate the surd.
- ✗Forgetting to simplify after rationalising. After clearing the surd from the denominator, check whether the numerator and denominator share common factors.
✦ Exam tips
- →Questions often combine simplifying surds with rationalising. Simplify the surds first, then rationalise.
- →When the denominator multiplies out to 1 (as in the worked example above), the answer simplifies very neatly. Watch for this — it can save time.
- →Write the conjugate multiplication step clearly. This earns a method mark even if you make an arithmetic error afterwards.