Rationalising binomial denominators
Objective. Use conjugates to rationalise denominators.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
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Learn: the key idea
Rationalising binomial denominators removes surds from a denominator of the form a+√b (or a−√b) by multiplying the top and bottom of the fraction by its conjugate, a−√b, since (a+√b)(a−√b) = a²−b is always rational. This turns the denominator into a whole number, leaving an equivalent fraction with no surd underneath. The numerator must be expanded fully after multiplying by the conjugate, and the final fraction simplified if possible. For example, to rationalise 1/(2+√3), multiply by (2−√3)/(2−√3) to get (2−√3)/(4−3) = 2−√3.
Worked example
Rationalise the denominator of 5/(3−√2), giving your answer in simplest form.
- 1Multiply top and bottom by the conjugate 3+√2
- 2Denominator becomes (3−√2)(3+√2) = 9−2 = 7
- 3Numerator becomes 5(3+√2) = 15+5√2
- 4Write as a single fraction: (15+5√2)/7
Answer: (15+5√2)/7
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Start practisingCommon mistake. Students sometimes multiply by the same expression rather than its conjugate, multiplying 1/(2+√3) by (2+√3)/(2+√3), which leaves a surd in the denominator instead of eliminating it; always flip the sign before the surd.
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