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Surds0%

Numeracy · Topic 3 of 6

Surds

Video lesson7 worked examples

Theory

Surds involve simplifying expressions and rationalising the denominators of fractions.

Exact values are an important method of communication in maths, science, and technology.

The key rules to memorise are:

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

\sqrt{\dfrac{a}{b}} = \dfrac{\sqrt{a}}{\sqrt{b}}

The Golden Rule: to simplify, split off the largest square factor. To rationalise, multiply top and bottom by the surd on the denominator. Both leave the value unchanged — you are only rewriting it in a tidier form.

⚠️ Common Examiner Traps

Not using the largest square factor: writing $\sqrt{72} = \sqrt{4}\sqrt{18} = 2\sqrt{18}$ is not finished — $\sqrt{18}$ still simplifies. Take out $36$ in one go.
Adding unlike surds: $\sqrt{2} + \sqrt{3}$ does not equal $\sqrt{5}$ . Only like surds combine, e.g. $2\sqrt{3} + 4\sqrt{3} = 6\sqrt{3}$ .
Splitting a sum under the root: $\sqrt{a+b} \neq \sqrt{a} + \sqrt{b}$ . The product rule works only for multiplication and division.
Stopping before simplifying: after rationalising, check whether the surd on top can still be reduced.

Worked examples

Example 1

Simplifying a Surd

Simplify $\sqrt{72}$ .

Step 1: Find the largest square number that is a factor of 72 (which is 36).

Step 2: Rewrite: $\sqrt{36 \times 2} = \sqrt{36} \times \sqrt{2}$ .

Answer: $6\sqrt{2}$ .

Example 2

Rationalising a Denominator

Rationalise $\frac{5}{\sqrt{3}}$ .

Step 1: Multiply the top and bottom by the surd in the denominator: $\frac{5 \times \sqrt{3}}{\sqrt{3} \times \sqrt{3}}$ .

Answer: $\frac{5\sqrt{3}}{3}$ .

Example 3

Collecting Like Surds

Simplify $\sqrt{48} + \sqrt{27}$ as a single surd.

Step 1: The two surds look different, so simplify each first by taking out its largest square factor: $\sqrt{48} = \sqrt{16 \times 3} = 4\sqrt{3}$ and $\sqrt{27} = \sqrt{9 \times 3} = 3\sqrt{3}$ .

Step 2: They are now like surds (both in $\sqrt{3}$ ), so add the numbers in front: $4\sqrt{3} + 3\sqrt{3} = 7\sqrt{3}$ .

Answer: $7\sqrt{3}$ .

Example 4

Multiplying Surds

Multiply and simplify $\sqrt{6} \times \sqrt{15}$ .

Step 1: Use $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$ to combine them under one root: $\sqrt{6 \times 15} = \sqrt{90}$ .

Step 2: Simplify by taking out the largest square factor of 90, which is 9: $\sqrt{90} = \sqrt{9 \times 10} = 3\sqrt{10}$ .

Answer: $3\sqrt{10}$ .

Example 5

Rationalise and Simplify

Express $\dfrac{8}{\sqrt{6}}$ with a rational denominator, giving your answer in its simplest form.

Step 1: Multiply top and bottom by $\sqrt{6}$ to clear the surd from the denominator: $\dfrac{8}{\sqrt{6}} \times \dfrac{\sqrt{6}}{\sqrt{6}} = \dfrac{8\sqrt{6}}{6}$ .

Step 2: The fraction is not yet in simplest form — $8$ and $6$ share a factor of 2:

\dfrac{8\sqrt{6}}{6} = \dfrac{4\sqrt{6}}{3}

Answer: $\dfrac{4\sqrt{6}}{3}$ .

Example 6

Expanding Surd Brackets

Expand and simplify $\sqrt{3}\left(\sqrt{6} + \sqrt{3}\right)$ .

Step 1: Multiply the outside surd by each term inside the bracket: $\sqrt{3} \times \sqrt{6} = \sqrt{18}$ and $\sqrt{3} \times \sqrt{3} = 3$ .

Step 2: Simplify the first surd — $\sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}$ :

\sqrt{3}\left(\sqrt{6} + \sqrt{3}\right) = 3\sqrt{2} + 3

Answer: $3\sqrt{2} + 3$ .

Example 7

🔗 Bringing it together

Simplify $\sqrt{50} + \sqrt{8} - \sqrt{18}$ .

Step 1: The three surds look unlike, so simplify each first by taking out its largest square factor: $\sqrt{50} = 5\sqrt{2}$ , $\sqrt{8} = 2\sqrt{2}$ , $\sqrt{18} = 3\sqrt{2}$ .

Step 2: They are now all like surds in $\sqrt{2}$ , so combine the numbers in front — watching the subtraction: $5\sqrt{2} + 2\sqrt{2} - 3\sqrt{2} = (5 + 2 - 3)\sqrt{2}$ .

Answer: $4\sqrt{2}$ .

Ready to practise?

12 Surds questions with answers and video solutions