Rationalising denominators
Objective. Rewrite simple surd fractions with rational denominators.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
Rationalising a denominator means rewriting a fraction so there is no surd left on the bottom, since dividing by an irrational number is considered unsimplified. For a simple surd denominator like a/√b, multiply both the numerator and denominator by √b, using the fact that √b × √b = b. For example, 3/√5 = (3 × √5)/(√5 × √5) = 3√5/5. This works because multiplying top and bottom by the same value does not change the fraction's value, only its form.
Worked example
Rationalise the denominator of 10/√2, giving your answer in its simplest form.
- 1Multiply top and bottom by √2: (10 × √2)/(√2 × √2).
- 2The denominator becomes √2 × √2 = 2.
- 3The numerator becomes 10√2.
- 4Simplify 10√2/2 = 5√2.
Answer: 5√2
Concept mastery
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Practice
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Start practisingCommon mistake. Pupils multiply only the denominator by √b and forget to multiply the numerator too, changing the value of the fraction. Always multiply both top and bottom by the same surd.
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