Surds Explained: Simplifying, Adding, Multiplying and Rationalising, with Examples (Years 9 to 10A)

Surds: how to simplify, add, multiply and rationalise them

Surds look harder than they are. Almost every surd question comes down to one move: pulling the largest square number out from under the root. Here’s each skill with a worked example, the mistakes that cost marks, and practice questions with answers.

By Ryan, founder of Bayside Academics, 15 years in private tutoring · Updated 6 November 2026

Quick answer

A surd is a root that can’t be simplified to a whole number, like √2. To simplify one, take out the largest square factor: √72 = √(36 × 2) = 6√2. You can only add like surds (3√5 + 2√5 = 5√5). Multiply by multiplying under the roots: √3 × √12 = √36 = 6. To rationalise, multiply the top and bottom by the surd in the denominator.

What a surd is

A surd is a root that can't be written as a whole number or fraction, like $\sqrt{2}$ , $\sqrt{5}$ or $\sqrt[3]{7}$ . Its decimal never ends or repeats ( $\sqrt{2} = 1.41421356\ldots$ ), so we keep it in surd form to stay exact.

$\sqrt{9}$ and $\sqrt{25}$ are not surds, because they equal 3 and 5.

Two rules do almost all the work:

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

\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}

Simplifying surds

Find the largest square number that goes into the number under the root: 4, 9, 16, 25, 36, 49, 64, 81, 100.

Worked example

Simplify $\sqrt{72}$ .

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

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

More: $\sqrt{50} = 5\sqrt{2}$ , $\sqrt{48} = 4\sqrt{3}$ , and $3\sqrt{20} = 3 \times 2\sqrt{5} = 6\sqrt{5}$ .

Adding and subtracting surds

Only like surds (the same number under the root) can be added or subtracted. Treat the surd like a letter: $3\sqrt{5} + 2\sqrt{5} = 5\sqrt{5}$ , just as $3x + 2x = 5x$ .

Simplify first, and unlike surds often become like ones:

Worked example

Simplify $\sqrt{12} + \sqrt{27}$ .

\sqrt{12} = 2\sqrt{3}, \qquad \sqrt{27} = 3\sqrt{3}

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

$\sqrt{2} + \sqrt{3}$ can't be simplified. Leave it as it is.

Multiplying and dividing surds

Multiply the numbers outside together and the numbers inside together, then simplify.

$\sqrt{3} \times \sqrt{12} = \sqrt{36} = 6$
$2\sqrt{5} \times 3\sqrt{2} = 6\sqrt{10}$
$\sqrt{7} \times \sqrt{7} = 7$ , so $(\sqrt{7})^2 = 7$
$\dfrac{\sqrt{50}}{\sqrt{2}} = \sqrt{25} = 5$

Expanding brackets with surds

Worked example

Expand $(\sqrt{5} + 2)(\sqrt{5} - 3)$ .

= 5 - 3\sqrt{5} + 2\sqrt{5} - 6

= -1 - \sqrt{5}

Difference of squares: $(\sqrt{3} + 1)(\sqrt{3} - 1) = 3 - 1 = 2$ . The surds disappear, which is the idea behind rationalising with a conjugate.

Rationalising the denominator

Multiply the top and bottom by the surd in the denominator, so the bottom becomes a whole number.

Worked example

Rationalise $\dfrac{6}{\sqrt{3}}$ .

\frac{6}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{6\sqrt{3}}{3}

= 2\sqrt{3}

Year 10A: using the conjugate

Rationalise $\dfrac{1}{\sqrt{3} + 1}$ . Multiply by the conjugate $\sqrt{3} - 1$ :

\frac{1}{\sqrt{3} + 1} \times \frac{\sqrt{3} - 1}{\sqrt{3} - 1}

= \frac{\sqrt{3} - 1}{3 - 1} = \frac{\sqrt{3} - 1}{2}

Common mistakes that cost marks

Splitting a sum. √(9 + 16) = √25 = 5, not √9 + √16 = 7.
Adding unlike surds. √2 + √3 is not √5.
Not simplifying fully. √72 = 2√18 isn’t finished: use the largest square, 36, to get 6√2.
Rationalising only the bottom. Whatever you multiply the denominator by, multiply the numerator by too.
Squaring a bracket term by term. (2 + √5)² = 9 + 4√5, not 4 + 5.

Practice questions with answers

Try each one before you open the answer.

Simplify $\sqrt{45}$
Show the answer
$\sqrt{9 \times 5} = 3\sqrt{5}$
Simplify $\sqrt{98}$
Show the answer
$\sqrt{49 \times 2} = 7\sqrt{2}$
Simplify $2\sqrt{18}$
Show the answer
$2 \times 3\sqrt{2} = 6\sqrt{2}$
Simplify $4\sqrt{3} + \sqrt{75}$
Show the answer
$\sqrt{75} = 5\sqrt{3}$ , so $4\sqrt{3} + 5\sqrt{3} = 9\sqrt{3}$
Simplify $\sqrt{8} \times \sqrt{6}$
Show the answer
$\sqrt{48} = \sqrt{16 \times 3} = 4\sqrt{3}$
Simplify $3\sqrt{2} \times 5\sqrt{6}$
Show the answer
$15\sqrt{12} = 15 \times 2\sqrt{3} = 30\sqrt{3}$
Simplify $\dfrac{\sqrt{80}}{\sqrt{5}}$
Show the answer
$\sqrt{\dfrac{80}{5}} = \sqrt{16} = 4$
Expand $\sqrt{3}(\sqrt{3} + 4)$
Show the answer
$3 + 4\sqrt{3}$
Expand $(2 + \sqrt{5})^2$
Show the answer
$4 + 4\sqrt{5} + 5 = 9 + 4\sqrt{5}$
Rationalise $\dfrac{8}{\sqrt{2}}$
Show the answer
$\dfrac{8\sqrt{2}}{2} = 4\sqrt{2}$
(Year 10A) Rationalise $\dfrac{4}{\sqrt{5} - 1}$
Show the answer
Multiply by the conjugate: $\dfrac{4(\sqrt{5} + 1)}{5 - 1} = \sqrt{5} + 1$

Surds by year level

Following the Victorian Curriculum 2.0. Schools sequence topics differently, so check your course.

Year 10A

Surds and fractional indices

Simplifying surds, operations, rationalising denominators and fractional indices, ready for senior Methods.

Worksheets and topic test · Free PDF with answers

Surds: common questions

What is a surd?

A surd is a root that can't be simplified to a whole number or fraction, like √2, √5 or ∛7. Its decimal goes on forever without repeating, so it's irrational. √9 isn't a surd, because it equals 3.

How do you simplify a surd?

Find the largest perfect square that divides the number under the root, split the root, and take the square root of that part. For example √72 = √(36 × 2) = 6√2.

Can you add √2 and √3?

No. You can only add or subtract like surds, which have the same number under the root. √2 + √3 stays as it is, but 3√5 + 2√5 = 5√5.

Why do we rationalise the denominator?

It's the convention for a final answer, and it makes surd answers easier to compare and add. Multiply the top and bottom by the surd in the denominator (or by the conjugate, in Year 10A).

What year level learns surds?

Students meet irrational numbers around Year 9. Working with surds (simplifying, operations and rationalising) is mainly a Year 10A topic in the Victorian Curriculum 2.0, ready for senior Methods maths.

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What a surd is

Simplifying surds

Adding and subtracting surds

Multiplying and dividing surds

Expanding brackets with surds

Rationalising the denominator

Common mistakes that cost marks

Practice questions with answers

Surds by year level

Surds: common questions

More maths help, Years 7 to 10

Got a surds question wrong in homework?