Surds: simplifying, rationalising the denominator and basic calculations

What you'll learn

This revision guide covers everything you need to know about surds for AQA GCSE Mathematics (Higher tier only). You'll learn how to recognise surds, simplify them using square factor rules, perform addition, subtraction and multiplication, and rationalise denominators. These skills are essential for problem-solving questions and appear regularly in Paper 1 (non-calculator) where exact answers are required.

Key terms and definitions

Surd — an irrational number that cannot be simplified to remove a square root sign, such as √2 or √5

Rational number — a number that can be written as a fraction p/q where p and q are integers (e.g. 0.5, 3, -2/7)

Irrational number — a number that cannot be written as a fraction; its decimal representation neither terminates nor recurs (e.g. √3, π)

Simplifying a surd — writing a surd in the form a√b where b has no square factors other than 1

Like surds — surds with the same irrational part that can be combined through addition or subtraction (e.g. 2√3 and 5√3)

Rationalising the denominator — the process of removing surds from the denominator of a fraction by multiplying by an appropriate form of 1

Conjugate — a pair of expressions that differ only in the sign between two terms (e.g. 3 + √2 and 3 - √2)

Exact form — an answer left as a surd rather than approximated as a decimal

Core concepts

Recognising surds

Not all square roots are surds. A square root is only a surd if it cannot be simplified to a rational number.

Square roots that ARE surds:

• √2, √3, √5, √6, √7, √8, √10, √11, etc.
• Any √n where n is not a perfect square

Square roots that are NOT surds:

• √4 = 2 (rational)
• √9 = 3 (rational)
• √16 = 4 (rational)
• √25 = 5 (rational)

At GCSE level, you must be able to identify whether an answer should remain in surd form or be simplified to a whole number. In non-calculator papers, questions often ask for answers in "exact form" or to "leave your answer as a surd", which means you must not use decimal approximations.

Simplifying surds

To simplify a surd, look for the largest square factor of the number under the root sign.

Key rule: √(a × b) = √a × √b

Method for simplifying:

1. Find the largest square factor of the number under the root
2. Split the surd into two factors using this square number
3. Simplify the square root of the perfect square
4. Write in the form a√b

Common square factors to look for:

• 4, 9, 16, 25, 36, 49, 64, 81, 100

Examples:

• √12 = √(4 × 3) = √4 × √3 = 2√3
• √18 = √(9 × 2) = √9 × √2 = 3√2
• √50 = √(25 × 2) = √25 × √2 = 5√2
• √48 = √(16 × 3) = √16 × √3 = 4√3
• √75 = √(25 × 3) = √25 × √3 = 5√3

When a surd includes a coefficient (number in front), simplify the surd part first, then multiply:

• 2√8 = 2 × √(4 × 2) = 2 × 2√2 = 4√2
• 3√20 = 3 × √(4 × 5) = 3 × 2√5 = 6√5

Adding and subtracting surds

You can only add or subtract like surds — those with the same number under the root sign.

Method:

1. Simplify all surds first
2. Identify like surds
3. Add or subtract the coefficients
4. Keep the surd part unchanged

Examples:

• 2√3 + 5√3 = 7√3
• 8√5 - 3√5 = 5√5
• √8 + √18 = 2√2 + 3√2 = 5√2
• 3√12 + √27 = 3 × 2√3 + 3√3 = 6√3 + 3√3 = 9√3
• √50 - √18 = 5√2 - 3√2 = 2√2

Important: You cannot combine unlike surds:

• 2√3 + 4√5 cannot be simplified further
• √2 + √7 remains as √2 + √7

Multiplying and dividing surds

Multiplying surds:

Use the rule: √a × √b = √(a × b)

Examples:

• √2 × √3 = √6
• √5 × √7 = √35
• √6 × √3 = √18 = 3√2
• 2√3 × 4√5 = 8√15
• 3√2 × 5√2 = 15 × 2 = 30

When multiplying a surd by itself: √a × √a = a

• √7 × √7 = 7
• √3 × √3 = 3

Expanding brackets with surds:

Use the same distributive law as with algebraic expressions.

Single bracket:

• 3(2 + √5) = 6 + 3√5
• √2(√8 + 3) = √16 + 3√2 = 4 + 3√2

Double brackets:

• (2 + √3)(4 + √3) = 8 + 2√3 + 4√3 + 3 = 11 + 6√3
• (1 + √5)(3 - √5) = 3 - √5 + 3√5 - 5 = -2 + 2√5
• (√7 + 2)(√7 - 2) = 7 - 2√7 + 2√7 - 4 = 3

Dividing surds:

Use the rule: √a ÷ √b = √(a ÷ b)

Examples:

• √12 ÷ √3 = √4 = 2
• √20 ÷ √5 = √4 = 2
• 8√6 ÷ 2√2 = 4√3

Rationalising the denominator (one term)

A fraction with a surd in the denominator is not considered to be in its simplest form at GCSE. You must rationalise the denominator.

Method for single-term denominators:

1. Multiply both numerator and denominator by the surd in the denominator
2. Simplify the resulting expression
3. Cancel common factors if possible

Examples:

Example 1: Rationalise 1/√2

Multiply by √2/√2: = (1 × √2)/(√2 × √2) = √2/2

Example 2: Rationalise 3/√5

Multiply by √5/√5: = (3 × √5)/(√5 × √5) = 3√5/5

Example 3: Rationalise 6/√3

Multiply by √3/√3: = (6 × √3)/(√3 × √3) = 6√3/3 = 2√3

Example 4: Rationalise 4/(2√7)

Multiply by √7/√7: = (4 × √7)/(2√7 × √7) = 4√7/(2 × 7) = 4√7/14 = 2√7/7

Rationalising the denominator (two terms)

When the denominator contains two terms (one or both involving surds), multiply by the conjugate.

Method:

1. Identify the conjugate of the denominator (change the sign between the terms)
2. Multiply numerator and denominator by this conjugate
3. Expand the brackets
4. Simplify

The denominator will always simplify because (a + b)(a - b) = a² - b²

Examples:

Example 1: Rationalise 1/(2 + √3)

Conjugate is (2 - √3)

= 1 × (2 - √3)/[(2 + √3)(2 - √3)] = (2 - √3)/(4 - 2√3 + 2√3 - 3) = (2 - √3)/1 = 2 - √3

Example 2: Rationalise 6/(√5 - 1)

Conjugate is (√5 + 1)

= 6(√5 + 1)/[(√5 - 1)(√5 + 1)] = (6√5 + 6)/(5 - 1) = (6√5 + 6)/4 = 3√5/2 + 3/2

Example 3: Rationalise (2 + √3)/(1 - √3)

Conjugate is (1 + √3)

= (2 + √3)(1 + √3)/[(1 - √3)(1 + √3)] = (2 + 2√3 + √3 + 3)/(1 - 3) = (5 + 3√3)/(-2) = -5/2 - 3√3/2

Worked examples

Example 1: Simplifying and combining surds (3 marks)

Question: Simplify fully: √45 + √20 - √5

Solution:

Step 1: Simplify each surd √45 = √(9 × 5) = 3√5 ✓ √20 = √(4 × 5) = 2√5 ✓

Step 2: Substitute and combine like surds 3√5 + 2√5 - √5 = 4√5 ✓

Answer: 4√5

Example 2: Expanding brackets with surds (3 marks)

Question: Expand and simplify (3 + √2)(5 - √2)

Solution:

Step 1: Multiply each term in the first bracket by each term in the second = 3 × 5 + 3 × (-√2) + √2 × 5 + √2 × (-√2) ✓

Step 2: Simplify each product = 15 - 3√2 + 5√2 - 2 ✓

Step 3: Collect like terms = 13 + 2√2 ✓

Answer: 13 + 2√2

Example 3: Rationalising with two terms (4 marks)

Question: Express 8/(3 - √5) in the form a + b√5 where a and b are integers.

Solution:

Step 1: Identify the conjugate of the denominator Conjugate of (3 - √5) is (3 + √5) ✓

Step 2: Multiply numerator and denominator by the conjugate = 8(3 + √5)/[(3 - √5)(3 + √5)] ✓

Step 3: Expand Numerator: 24 + 8√5 Denominator: 9 - 5 = 4 ✓

Step 4: Simplify = (24 + 8√5)/4 = 6 + 2√5 ✓

Answer: 6 + 2√5 (so a = 6, b = 2)

Common mistakes and how to avoid them

• Incorrectly adding unlike surds: Students often write √2 + √3 = √5, which is wrong. You can only add surds with the same number under the root. Always simplify first, then check if the surds are like surds before combining.

• Forgetting to simplify surds fully: Writing √12 when the answer should be 2√3. Always check whether the number under the root has any square factors (4, 9, 16, 25, 36, etc.) and simplify completely.

• Making errors when squaring surds: Remember that √a × √a = a, not 2a. For example, √5 × √5 = 5, not 10.

• Leaving surds in the denominator: At GCSE, fractions must have rational denominators. Always rationalise expressions like 3/√2 by multiplying top and bottom by the surd.

• Sign errors when rationalising two-term denominators: When multiplying by the conjugate, be careful with negative signs. (a + b)(a - b) = a² - b², not a² + b².

• Giving decimal approximations in exact form questions: If a question asks for an exact answer or says "leave your answer in surd form", do not round to decimals. √2 is exact; 1.414 is not.

Exam technique for surds

• Command words matter: "Simplify" means write in simplest surd form with no square factors under the root. "Rationalise" specifically means remove surds from the denominator. "Show that" requires you to demonstrate every step clearly, arriving at the given answer.

• Non-calculator papers: Surds appear almost exclusively on Paper 1 (non-calculator). If you're reaching for a calculator and see square roots, you're likely meant to work with exact surd form instead.

• Method marks are available: Even if your final answer is wrong, you can gain marks for correct methods such as identifying the conjugate, expanding brackets correctly, or simplifying individual surds. Always show your working clearly.

• Check your final answer: Is it fully simplified? Have you rationalised if required? Does it match the form requested in the question (e.g. a + b√c)? Taking 10 seconds to verify can secure those final marks.

Quick revision summary

Surds are irrational square roots that cannot be simplified to whole numbers. Simplify surds by finding the largest square factor. Only combine like surds through addition/subtraction. Multiply surds using √a × √b = √(ab). Rationalise denominators by multiplying by the surd (single term) or the conjugate (two terms). Always leave answers in exact form on non-calculator papers. Watch for square factors: 4, 9, 16, 25, 36, 49, 64, 81, 100. Show all working for method marks.