How to Simplify Surds and Rationalise

📋 Spec coverage: GCSE AQA Mathematics (8300) This article covers AQA N8 (Higher): calculating exactly with surds, simplifying surd expressions involving square factors, and rationalising denominators.

To simplify a surd, take square factors outside the root. To rationalise a denominator, multiply by an expression that removes the surd from the denominator.

What is a surd?

A surd is an irrational root written exactly, such as $\sqrt{3}$. It cannot be written as a terminating or recurring decimal, so keeping it as a surd preserves its exact value.

Use the rule:

$$\sqrt{ab}=\sqrt{a}\sqrt{b}$$

Simplifying and calculating with surds

Simplify $\sqrt{72}$ by finding its largest square factor:

$$\sqrt{72}=\sqrt{36\times 2}=\sqrt{36}\sqrt{2}=6\sqrt{2}$$

Choosing the largest square factor avoids repeating the process. Surds with the same irrational part are like surds, so their coefficients can be combined:

$$3\sqrt{2}+5\sqrt{2}=8\sqrt{2}$$

However, $3\sqrt{2}+5\sqrt{3}$ cannot be simplified further because the irrational parts differ.

Multiplication can also be simplified exactly:

$$\sqrt{3}\times 2\sqrt{6}=2\sqrt{18}=6\sqrt{2}$$

Always simplify the final surd and leave the answer exact unless the question asks for a decimal.

Rationalising a denominator

A rational denominator contains no surd. Rationalise:

$$\frac{5}{\sqrt{8}}$$

First, simplify $\sqrt{8}=2\sqrt{2}$. Then multiply the numerator and denominator by $\sqrt{2}$:

$$\frac{5}{2\sqrt{2}}\times \frac{\sqrt{2}}{\sqrt{2}}=\frac{5\sqrt{2}}{2\times 2}=\frac{5\sqrt{2}}{4}$$

The fraction is unchanged because $\frac{\sqrt{2}}{\sqrt{2}}=1$.

For a two-term denominator, multiply by its conjugate, which has the opposite sign. The conjugate of $2+\sqrt{3}$ is $2-\sqrt{3}$. This works because the middle terms cancel:

$$(2+\sqrt{3})(2-\sqrt{3})=4-3=1$$

For example:

$$\frac{1}{2+\sqrt{3}}\times \frac{2-\sqrt{3}}{2-\sqrt{3}}=2-\sqrt{3}$$

Using the conjugate creates a difference of two squares, so the denominator becomes rational.

Common exam mistake

Do not assume $\sqrt{a+b}=\sqrt{a}+\sqrt{b}$. For example, $\sqrt{9+16}=5$, not $3+4=7$. Also rationalise fully: no surd should remain in the denominator.

A frequent sign error is multiplying by the same binomial instead of the conjugate. Show each stage clearly so a correct method is visible even if you make an arithmetic slip.

Completeness check: All listed AQA N8 Higher skills are covered: exact surd calculation, simplifying using square factors, and rationalising single-term and binomial denominators.