🔘 Rationalising the Denominator

About Rationalising the Denominator

A fraction with a surd on the bottom is not wrong, but it is not finished, and rationalising it is one of those conventions that only makes sense once you have tried to compare two such fractions by eye. The method is short — multiply top and bottom by the surd — and the marks are lost in the tidying up afterwards, where the radicand and the new denominator often share a factor that should be cancelled. This worksheet targets exactly that. Every denominator is genuinely irrational before rationalising: the radicands are all non-square, so a question whose denominator is already rational never appears. The four options are the errors pupils actually make rather than random expressions — multiplying the top only and leaving the surd downstairs, dividing by the radicand instead of multiplying, and failing to reduce the result. The keyed answer always has a rational denominator, which is the property being tested, and it is computed from a structured surd representation rather than assembled as text. Numerators and radicands grow across the five levels, so the cancelling step becomes less obvious as the numbers get larger. Eight questions per page with tick boxes and its own separate answer key. Its on-screen twin is Rationalise Denominator Quiz.

Common questions

Is every denominator actually irrational to start with?

Yes. All the radicands are non-square, so a question whose denominator is already rational never appears.

Where do pupils usually lose marks?

In the tidying up — the radicand and the new denominator often share a factor that should be cancelled.

How are the wrong options chosen?

They are the real errors: multiplying the top only, dividing instead of multiplying, and failing to reduce.

Is an answer key included?

Yes. Every level downloads with a key giving the letter and the simplified expression.