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Number · Surds

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Rationalising the denominator

Why multiplying top and bottom by the same thing changes nothing, how √a in a denominator is cleared by √a, and how a conjugate clears a two-term denominator like 3 + √5.

Number · Surds

Rationalising the denominator

Why multiplying top and bottom by the same thing changes nothing, how √a in a denominator is cleared by √a, and how a conjugate clears a two-term denominator like 3 + √5.

Why it works

"Rationalise the denominator" means: rewrite the fraction so no surd is left underneath. The tool is the most innocent fact in fractions: multiplying top and bottom by the same number multiplies the fraction by 11 — the value doesn't change, only its appearance. (Multiplying only the bottom is not allowed: that genuinely changes the number.)

One-term denominators: multiply by the surd itself. For 63\frac{6}{\sqrt{3}}, multiply top and bottom by 3\sqrt{3}:

63=6333=633=23.\frac{6}{\sqrt{3}} = \frac{6\sqrt{3}}{\sqrt{3}\sqrt{3}} = \frac{6\sqrt{3}}{3} = 2\sqrt{3}.

The bottom worked because 33=3\sqrt{3}\sqrt{3} = 3 exactly. Always finish by simplifying — 633\frac{6\sqrt{3}}{3} earns the method mark, but the final mark wants 232\sqrt{3}.

Two-term denominators: multiply by the conjugate. Multiplying 3+53 + \sqrt{5} by itself does not clear the surd — (3+5)2(3 + \sqrt{5})^2 still contains 656\sqrt{5}. What clears it is the conjugate, the same pair with the opposite sign, because conjugates make a difference of two squares:

(3+5)(3−5)=9−5=4.(3 + \sqrt{5})(3 - \sqrt{5}) = 9 - 5 = 4.

So for 43+5\frac{4}{3 + \sqrt{5}}, multiply top and bottom by 3−53 - \sqrt{5}:

43+5=4(3−5)(3+5)(3−5)=12−454=3−5.\frac{4}{3 + \sqrt{5}} = \frac{4(3 - \sqrt{5})}{(3 + \sqrt{5})(3 - \sqrt{5})} = \frac{12 - 4\sqrt{5}}{4} = 3 - \sqrt{5}.

Why bother? Partly convention, but mostly usefulness: a rational denominator makes fractions comparable and addable, and in the exam the target form "a+bca + b\sqrt{c} where aa and bb are integers" forces the conjugate method — you cannot reach that form without it. Expect the answer to collapse neatly; these questions are engineered so the denominator becomes a small whole number. If yours becomes 00 or something ugly, check the conjugate's sign.

Sense-check exact answers. 63=23\frac{6}{\sqrt{3}} = 2\sqrt{3}: as decimals, 6÷1.732…≈3.466 \div 1.732\ldots \approx 3.46 and 2×1.732…≈3.462 \times 1.732\ldots \approx 3.46. ✓ A ten-second decimal check catches most slips.