Surds [H]

This lesson covers surds — irrational numbers that are left in root form — for AQA GCSE Mathematics Higher Tier. Surds provide exact values where decimals would be rounded approximations. You need to be able to simplify surds, perform operations with them, and rationalise denominators.

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What Is a Surd?

A surd is a root that cannot be simplified to a whole number or a fraction. It is an irrational number — its decimal form goes on forever without repeating.

Expression	Surd?	Reason
Square root of 2	Yes	1.41421356... (irrational)
Square root of 9	No	Equals 3 (rational)
Square root of 5	Yes	2.23606797... (irrational)
Square root of 100	No	Equals 10 (rational)
Cube root of 7	Yes	1.91293118... (irrational)
Square root of 16/25	No	Equals 4/5 (rational)

Exam Tip: The question will usually say "give your answer in surd form" or "give an exact answer". This means do NOT use a calculator to find a decimal — leave roots in your answer.

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Simplifying Surds

To simplify a surd, find the largest perfect square factor of the number under the root.

The Key Rule

Square root of (a x b) = square root of a x square root of b

Worked Example 1

Simplify the square root of 72.

Step 1: Find the largest perfect square factor of 72.

• 72 = 36 x 2
• 36 is a perfect square (6 squared)

Step 2: Split the root:

• Square root of 72 = square root of (36 x 2) = square root of 36 x square root of 2

Step 3: Simplify:

• = 6 x square root of 2

Answer: 6 root 2

Worked Example 2

Simplify the square root of 200.

• 200 = 100 x 2
• Square root of 200 = square root of 100 x square root of 2 = 10 root 2

Worked Example 3

Simplify the square root of 48.

• 48 = 16 x 3
• Square root of 48 = square root of 16 x square root of 3 = 4 root 3

Perfect Squares to Know

Number	Square
1	1
2	4
3	9
4	16
5	25
6	36
7	49
8	64
9	81
10	100
11	121
12	144

Exam Tip: Always look for the LARGEST perfect square factor. Using smaller factors means more steps. For example, square root of 72 = square root of (4 x 18) = 2 root 18, but root 18 can be simplified further. Starting with 36 x 2 gives you 6 root 2 in one step.

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Adding and Subtracting Surds

You can only add or subtract surds that have the same number under the root (like terms).

Examples

• 3 root 5 + 7 root 5 = 10 root 5 (add the coefficients)
• 8 root 3 - 2 root 3 = 6 root 3 (subtract the coefficients)
• 5 root 2 + 3 root 7 = 5 root 2 + 3 root 7 (cannot simplify — different surds)

Worked Example: Simplify First

Simplify 2 root 12 + 3 root 3.

Step 1: Simplify root 12:

• Root 12 = root (4 x 3) = 2 root 3

Step 2: Substitute:

• 2 x 2 root 3 + 3 root 3 = 4 root 3 + 3 root 3

Step 3: Add:

• = 7 root 3

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Multiplying Surds

Rule

Root a x root b = root (a x b)

Worked Examples

• Root 3 x root 5 = root 15
• Root 2 x root 8 = root 16 = 4
• 3 root 2 x 4 root 5 = (3 x 4) x root (2 x 5) = 12 root 10

Important Special Case

Root a x root a = a

This is because root a squared = a.

Examples:

• Root 5 x root 5 = 5
• Root 11 x root 11 = 11

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Expanding Brackets with Surds

Use the same expansion techniques as algebra.

Single Bracket

Expand root 3(2 + root 3).

= root 3 x 2 + root 3 x root 3 = 2 root 3 + 3

Answer: 3 + 2 root 3

Double Brackets

Expand (2 + root 5)(3 - root 5).

Use FOIL:

• First: 2 x 3 = 6
• Outer: 2 x (-root 5) = -2 root 5
• Inner: root 5 x 3 = 3 root 5
• Last: root 5 x (-root 5) = -5

Combine: 6 - 2 root 5 + 3 root 5 - 5 = 1 + root 5

Difference of Two Squares

(a + root b)(a - root b) = a squared - b

This is particularly important for rationalising denominators.

Example: (3 + root 2)(3 - root 2) = 9 - 2 = 7

Exam Tip: The difference of two squares pattern eliminates the surd entirely. This is the basis for rationalising denominators with two terms.

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Dividing Surds

Rule

Root a / root b = root (a / b)

Worked Example

Root 20 / root 5 = root (20/5) = root 4 = 2

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Rationalising the Denominator

Rationalising the denominator means rewriting a fraction so that there is no surd in the denominator.

Type 1: Single Surd in Denominator

Multiply top and bottom by the surd.

Rationalise 5 / root 3.

= (5 x root 3) / (root 3 x root 3) = 5 root 3 / 3

Answer: (5 root 3) / 3