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Surds

A surd is a root that cannot be written as an exact decimal — like √2 or √5. Working in surd form keeps answers exact, which is exactly what A-Level demands. This lesson covers simplifying surds and the three rules for manipulating them.

30 min Video by Zeeshan Zamurred Algebraic Expressions

Edexcel AS Level Maths: 1.5 SurdsWatch the full walkthrough before the notes below.

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What you'll be able to do

Understand what a surd is and why we keep exact form
Use the multiplication and division rules for surds
Simplify a surd by extracting square factors
Add and subtract like surds
Expand brackets that contain surds

What is a surd?

A $surd$ is an irrational root — a root whose value is a never-ending, non-repeating decimal. For example $2 = 1.41421356 \dots$ , so we leave it as $2$ to stay exact.

Roots like $9 = 3$ are $not$ surds, because they work out to a whole number.

Tip — If a question says "give your answer in surd form" or "exact value", do not reach for the calculator decimal — keep the root.

The rules for multiplying and dividing surds

Surds combine under multiplication and division by combining what is inside the roots. These two rules are the engine behind everything else in the topic.

a \times b = ab

Multiply the insides under one root.

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

Divide the insides under one root.

1Combine under one root:

3 \times 12 = 36

36 = 6

Answer

6

Simplifying surds

To simplify a surd, find the $largest square number$ that divides into the value, split it off using the multiplication rule, and take its root.

ab = a b

Split off a perfect-square factor, then root it.

1The largest square factor of

50

25

(

50 = 25 \times 2

50 = 252 = 52

Answer

52

Tip — Use the LARGEST square factor. Using a smaller one (e.g. 4 instead of 36 for √72) means you have to simplify again.

Adding, subtracting and expanding

You can only add or subtract $like$ surds — surds with the same number under the root — just like collecting like terms. Sometimes you must simplify first before they match.

Brackets with surds expand exactly like algebraic brackets; remember that $a \times a = a$ .

1Simplify

18 = 92 = 32

2Now they are like surds:

32 + 2 = 42

Answer

42

Tip — √a × √a = a. This single fact removes most surds when you expand brackets.

Formula recap

a \times b = ab

Multiplying surds.

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

Dividing surds.

a \times a = a

A surd times itself removes the root.

50 = 52

Simplify by extracting a square factor.

Common mistakes to avoid

Writing √a + √b = √(a + b), e.g. √9 + √16 = √25 = 5.

Addition does NOT combine under the root: √9 + √16 = 3 + 4 = 7.

Adding unlike surds, e.g. 2√3 + 3√2 = 5√5.

Only like surds combine. 2√3 + 3√2 cannot be simplified further.

Using a small square factor and stopping, e.g. √72 = 2√18.

Use the largest square factor: √72 = 6√2.

Key takeaways

A surd is an irrational root; keep it exact rather than rounding.
√a × √b = √(ab) and √a ÷ √b = √(a/b) — but this never works for + or −.
Simplify by splitting off the largest perfect-square factor.
Add/subtract only like surds; √a × √a = a clears roots when expanding.

Test yourself

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