Rationalising the Denominator

Grade 8

Why can't we leave a surd on the bottom?

In maths, a fraction with a surd in the denominator is not in its simplest form. The convention is to always write fractions with a rational denominator — meaning a whole number or a fraction, not an irrational root.

This is called rationalising the denominator. It makes the expression easier to work with and is what exam mark schemes expect as a final answer.

Rationalising does not change the value of the fraction. You are simply rewriting it in an equivalent, tidier form.

The method — multiply top and bottom by the surd

The trick is to multiply the fraction by a form of 1. If the denominator is √a, multiply the top and the bottom by √a. The denominator becomes √a × √a = a, which is a rational number.

1 ÷ √a = (1 × √a) ÷ (√a × √a) = √a ÷ a

Because you are multiplying by √a ÷ √a (which equals 1), the value of the fraction stays exactly the same — you have only changed its form.

Worked example

EXAMPLE

Rationalise the denominator of 5 ÷ √3. Give your answer in simplest form.

Multiply top and bottom by √3: (5 × √3) ÷ (√3 × √3) = 5√3 ÷ 3. There are no common factors to cancel and the denominator is now rational, so the answer is 5√3 ÷ 3. Written as a fraction: 5√3 over 3.

EXAMPLE

Rationalise the denominator of 6 ÷ √8. Simplify fully.

First simplify √8: √8 = √(4 × 2) = 2√2. So the fraction becomes 6 ÷ 2√2 = 3 ÷ √2. Now rationalise: multiply top and bottom by √2: (3 × √2) ÷ (√2 × √2) = 3√2 ÷ 2. The answer is 3√2 ÷ 2.

Watch out

Always simplify the surd in the denominator before rationalising — it keeps the numbers smaller and reduces errors. If you rationalise √8 directly you get 6√8 ÷ 8 and then have to simplify at the end, which is more work and more chance of a slip.

Try this one

Work out \( 3\sqrt{2} \times 4\sqrt{2} \). Give your answer as an integer. [2]

Nova's hint:

This is about multiplying surds together using the rule \( \sqrt{a} \times \sqrt{a} = a \).

Ask yourself: 'what happens when I multiply the whole-number parts together and separately multiply the surd parts?' Try multiplying the coefficients first, then the surds.

Try to Solve it with Nova?