Learn | Grasp Maths

Lesson overview

Number - surds and exact values

A surd is the root of a number that cannot be simplified to remove the radical sign. Surds allow us to express exact values rather than decimals. For example, $\sqrt{2}$ is a surd, but $\sqrt{4} = 2$ is not. We can simplify surds using the product rule: $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$ . We can also rationalise the denominator to remove surds from the bottom of a fraction: $\frac{1}{\sqrt{n}} = \frac{\sqrt{n}}{n}$ . Working with surds is essential in GCSE maths because many exact answers are expressed as surds.

Simplifying surds

Simplify $\sqrt{12}$ and $\sqrt{18}$ .

Find the largest square number that divides the number under the surd. Use the product rule to split it.

Multiplying surds

Calculate $\sqrt{2} \times \sqrt{8}$ .

Multiply under the surd sign, then simplify the result.

Rationalising the denominator

Rationalise $\frac{1}{\sqrt{3}}$ .

Multiply numerator and denominator by the surd to remove it from the denominator.

Expanding brackets with surds

Expand $(2 + \sqrt{3})(1 + \sqrt{3})$ .

Use FOIL (or distribution) carefully. Remember that $\sqrt{a} \times \sqrt{a} = a$ .

Worked example

Simplify $\frac{4}{\sqrt{2}}$ and express in the form $a\sqrt{b}$ where $a$ and $b$ are integers.

Identify that the denominator is a surd: $\sqrt{2}$ .
Multiply numerator and denominator by $\sqrt{2}$ to rationalise.
The numerator becomes $4 \times \sqrt{2} = 4\sqrt{2}$ .
The denominator becomes $\sqrt{2} \times \sqrt{2} = 2$ .
Simplify the fraction: $\frac{4\sqrt{2}}{2} = 2\sqrt{2}$ .
The answer is $2\sqrt{2}$ , where $a = 2$ and $b = 2$ .

Try it

Work through each question carefully. Check that you have simplified surds fully and rationalised denominators.

Question 1

Simplify $\sqrt{20}$ .

💡 Find the largest square number that divides 20.

Question 2

Simplify $\sqrt{8} + \sqrt{2}$ .

💡 Simplify $\sqrt{8}$ first using the product rule, then add like surds.

Question 3

Rationalise $\frac{2}{\sqrt{5}}$ .

💡 Multiply numerator and denominator by $\sqrt{5}$ .

Question 4

Expand and simplify $(\sqrt{2} + 1)^2$ .

💡 Use $(a + b)^2 = a^2 + 2ab + b^2$ . Remember $\sqrt{2} \times \sqrt{2} = 2$ .

Question 5

Simplify $\frac{3\sqrt{8}}{\sqrt{2}}$ .

💡 Simplify $\sqrt{8}$ first, then divide by $\sqrt{2}$ . Alternatively, use $\frac{\sqrt{8}}{\sqrt{2}} = \sqrt{\frac{8}{2}} = \sqrt{4}$ .

Common mistakes

Watch for these when working through the lesson.

Forgetting to simplify surds fully. For example, leaving $\sqrt{12}$ instead of simplifying to $2\sqrt{3}$ .
Adding surds that are not like terms. For example, $\sqrt{2} + \sqrt{3} \neq \sqrt{5}$ . Only combine like surds such as $2\sqrt{3} + 5\sqrt{3} = 7\sqrt{3}$ .
Not rationalising the denominator. Always move surds from the bottom of a fraction to the top.
Expanding brackets incorrectly. Remember to use FOIL and be careful with $\sqrt{a} \times \sqrt{a} = a$ , not $2a$ .

Surds and exact form

Lesson overview

Worked example

Try it

Common mistakes

Related topics

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