Surds& Rationalising the Denominator.

Q: What is a surd in IGCSE Maths?

A surd is an irrational root that cannot be simplified to a whole number or exact fraction. Examples include √2, √3, √5, and 3√7. Numbers like √4 (= 2) or √9 (= 3) are NOT surds because they simplify to rational numbers. Surds are left in root form to give exact answers rather than rounded decimals.

Q: How do you know when a surd is fully simplified?

A surd √n is fully simplified when the number under the root sign has no perfect square factors other than 1. To check, try dividing by perfect squares (4, 9, 16, 25, 36, ...). For example, √12 is not simplified because 12 = 4 × 3, so √12 = 2√3. But √15 is already simplified because 15 has no perfect square factors.

Q: What is the conjugate method for rationalising?

When the denominator contains a surd expression like (a + √b), you multiply both the numerator and denominator by the conjugate (a − √b). This works because (a + √b)(a − √b) = a² − b, which eliminates the surd from the denominator. The same principle applies if the denominator is (a − √b) — you multiply by (a + √b).

Q: Are surds on both IGCSE papers?

Surds are a Higher tier topic and can appear on both Paper 1 (non-calculator) and Paper 2 (calculator) of the IGCSE Edexcel Maths exam. However, they are more commonly tested on Paper 1 because the whole point of surds is to work with exact values without a calculator. Expect 3-6 marks on surd questions across the papers.

Higher tier essential: simplify, add, subtract, multiply surds and rationalise denominators using the conjugate method.

Hire a Tutor

6+ Years

Full-Time Professional Educators

In-Person Only

2,000+ Families

What are surds and how do you rationalise the denominator in IGCSE Maths?

Surds are irrational roots that cannot be simplified to whole numbers (e.g. √2, √5, 3√7). Rationalising the denominator means eliminating the surd from the bottom of a fraction. For a single surd denominator, multiply top and bottom by the surd. For a denominator like (a + √b), multiply by the conjugate (a − √b). This Higher-tier topic typically carries 3-6 marks on the IGCSE paper.

Tier: Higher OnlyMarks: 3-6Key Skill: Conjugate Method

Source: Edexcel IGCSE Mathematics (9-1) Specification 4MA1

Key Takeaway

To simplify a surd like √72, find the largest perfect square factor: √72 = √(36 × 2) = 6√2. To rationalise a/(b + √c), multiply numerator and denominator by the conjugate (b − √c). The denominator becomes b² − c, eliminating the surd.

How do you simplify surds?

Simplifying a surd means rewriting it so the number under the root sign is as small as possible. The method relies on the rule:

√(a × b) = √a × √b

To simplify √n, find the largest perfect square that divides n. Perfect squares to know: 4, 9, 16, 25, 36, 49, 64, 81, 100.

√50: 50 = 25 × 2 → √50 = √25 × √2 = 5√2
√72: 72 = 36 × 2 → √72 = √36 × √2 = 6√2
√48: 48 = 16 × 3 → √48 = √16 × √3 = 4√3

Always use the largest perfect square factor. Using a smaller one (e.g. √72 = √4 × √18 = 2√18) gives a correct but not fully simplified result — you would need to simplify again.

How do you add, subtract and multiply surds?

Adding and subtracting

You can only add or subtract surds that have the same number under the root (like terms). Simplify each surd first, then combine:

3√5 + 2√5 = 5√5
√50 + √18 = 5√2 + 3√2 = 8√2
√12 − √3 = 2√3 − √3 = √3

You cannot add unlike surds: √2 + √3 stays as √2 + √3.

Multiplying surds

Use the rule √a × √b = √(ab). For expressions like (2 + √3)(4 − √3), expand using FOIL just as with algebraic brackets:

First: 2 × 4 = 8
Outer: 2 × (−√3) = −2√3
Inner: √3 × 4 = 4√3
Last: √3 × (−√3) = −3
Combine: 8 − 2√3 + 4√3 − 3 = 5 + 2√3

How do you rationalise a single surd denominator?

When the denominator is a single surd (e.g. 1/√3), multiply both top and bottom by that surd:

1/√3 × √3/√3 = √3/3

This works because √3 × √3 = 3, removing the surd from the denominator. The value of the fraction is unchanged because you are multiplying by 1 (in the form √3/√3).

For a denominator like 2√5, multiply by √5/√5. The denominator becomes 2 × 5 = 10.

How do you rationalise using the conjugate?

When the denominator is a two-term expression containing a surd — such as (3 + √2) or (5 − √7) — you use the conjugate. The conjugate of (a + √b) is (a − √b), and vice versa.

The key identity is the difference of two squares:

(a + √b)(a − √b) = a² − b

This always produces a rational denominator. For example, to rationalise 5/(3 + √2):

Multiply top and bottom by (3 − √2)
Numerator: 5(3 − √2) = 15 − 5√2
Denominator: (3 + √2)(3 − √2) = 9 − 2 = 7
Result: (15 − 5√2)/7

Worked Examples

Grade 7

Simplify √50 + √18. Give your answer in the form a√2.

Step 1: √50 = √(25 × 2) = 5√2

Step 2: √18 = √(9 × 2) = 3√2

Step 3: 5√2 + 3√2 = 8√2

Answer: 8√2

Grade 8

Rationalise the denominator: 1/√3. Give your answer in its simplest form.

Step 1: Multiply numerator and denominator by √3.

Step 2: Numerator becomes 1 × √3 = √3

Step 3: Denominator becomes √3 × √3 = 3

Answer: √3/3

Grade 8

Expand and simplify (2 + √5)(3 − √5).

Step 1 — First: 2 × 3 = 6

Step 2 — Outer: 2 × (−√5) = −2√5

Step 3 — Inner: √5 × 3 = 3√5

Step 4 — Last: √5 × (−√5) = −5

Step 5 — Collect terms: 6 − 2√5 + 3√5 − 5 = 1 + √5

Answer: 1 + √5

Grade 9

Rationalise fully: 5/(3 + √2). Give your answer in the form a + b√2.

Step 1 — Identify conjugate: The conjugate of (3 + √2) is (3 − √2).

Step 2 — Multiply top and bottom: [5(3 − √2)] / [(3 + √2)(3 − √2)]

Step 3 — Expand numerator: 5 × 3 − 5 × √2 = 15 − 5√2

Step 4 — Expand denominator: 3² − (√2)² = 9 − 2 = 7

Step 5 — Write result: (15 − 5√2)/7

Answer: (15 − 5√2)/7 or equivalently 15/7 − (5/7)√2

Common Mistakes With Surds

Adding unlike surds: Writing √2 + √3 = √5 is wrong. You can only combine surds with the same radicand.

Not fully simplifying: Leaving √12 as 2√3 is correct, but leaving it as √12 loses the mark. Always look for perfect square factors.

Forgetting to multiply both top and bottom: When rationalising, you must multiply the numerator AND denominator by the same expression, or you change the value of the fraction.

Sign error in the conjugate: The conjugate of (a + √b) is (a − √b), not (−a + √b). Only the sign of the surd term changes.

Exam Tips

Memorise perfect squares up to 144: Quick recognition of 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144 speeds up surd simplification significantly.

Use FOIL for bracket expansion: Treat surd expansions exactly like algebraic bracket expansion. Write all four terms before collecting like terms.

Show every step when rationalising: Examiners award method marks for identifying the conjugate, expanding correctly, and simplifying. Do not skip steps.

Link surds to indices: Remember that √a = a^(1/2). This connection to fractional indices is often tested in Higher tier questions.

Number System Hub

All number topics in one place

Worksheets & Answers

Free practice for number topics

Laws of Indices

Negative & fractional indices

Formula Sheet

Visual guide to key formulae

IGCSE Tutors in Dubai

Specialist in-home IGCSE support

Maths Tutors in Dubai

Expert maths tutoring at home

Contact GetYourTutors — IGCSE Maths Surds

Phone: (+971) 4-313-2715 | Mobile: 050-947-9432

WhatsApp: 050-947-9432

Email: info@getyourtutors.com

Emirates Towers, Office Tower, Level 41, Sheikh Zayed Road, PO Box 31003, Dubai, UAE

Last updated: March 2026

Frequently Asked Questions

Everything you need to know about our private tutoring services in Dubai.

What is a surd in IGCSE Maths?

How do you know when a surd is fully simplified?

What is the conjugate method for rationalising?

Are surds on both IGCSE papers?

Struggling With Higher Tier IGCSE Maths?

A dedicated tutor builds confidence in challenging topics like surds. In-home across Dubai.

Hire a Tutor WhatsApp Us

Loading…

How do you simplify surds?

Simplifying a surd means rewriting it so the number under the root sign is as small as possible. The method relies on the rule:

√(a × b) = √a × √b

To simplify √n, find the largest perfect square that divides n. Perfect squares to know: 4, 9, 16, 25, 36, 49, 64, 81, 100.

√50: 50 = 25 × 2 → √50 = √25 × √2 = 5√2
√72: 72 = 36 × 2 → √72 = √36 × √2 = 6√2
√48: 48 = 16 × 3 → √48 = √16 × √3 = 4√3

Always use the largest perfect square factor. Using a smaller one (e.g. √72 = √4 × √18 = 2√18) gives a correct but not fully simplified result — you would need to simplify again.

How do you add, subtract and multiply surds?

Adding and subtracting

You can only add or subtract surds that have the same number under the root (like terms). Simplify each surd first, then combine:

3√5 + 2√5 = 5√5
√50 + √18 = 5√2 + 3√2 = 8√2
√12 − √3 = 2√3 − √3 = √3

You cannot add unlike surds: √2 + √3 stays as √2 + √3.

Multiplying surds

Use the rule √a × √b = √(ab). For expressions like (2 + √3)(4 − √3), expand using FOIL just as with algebraic brackets:

First: 2 × 4 = 8
Outer: 2 × (−√3) = −2√3
Inner: √3 × 4 = 4√3
Last: √3 × (−√3) = −3
Combine: 8 − 2√3 + 4√3 − 3 = 5 + 2√3

How do you rationalise a single surd denominator?

When the denominator is a single surd (e.g. 1/√3), multiply both top and bottom by that surd:

1/√3 × √3/√3 = √3/3

This works because √3 × √3 = 3, removing the surd from the denominator. The value of the fraction is unchanged because you are multiplying by 1 (in the form √3/√3).

For a denominator like 2√5, multiply by √5/√5. The denominator becomes 2 × 5 = 10.

How do you rationalise using the conjugate?

When the denominator is a two-term expression containing a surd — such as (3 + √2) or (5 − √7) — you use the conjugate. The conjugate of (a + √b) is (a − √b), and vice versa.

The key identity is the difference of two squares:

(a + √b)(a − √b) = a² − b

This always produces a rational denominator. For example, to rationalise 5/(3 + √2):

Multiply top and bottom by (3 − √2)
Numerator: 5(3 − √2) = 15 − 5√2
Denominator: (3 + √2)(3 − √2) = 9 − 2 = 7
Result: (15 − 5√2)/7

Worked Examples

Grade 7

Simplify √50 + √18. Give your answer in the form a√2.

Step 1: √50 = √(25 × 2) = 5√2

Step 2: √18 = √(9 × 2) = 3√2

Step 3: 5√2 + 3√2 = 8√2

Answer: 8√2

Grade 8

Rationalise the denominator: 1/√3. Give your answer in its simplest form.

Step 1: Multiply numerator and denominator by √3.

Step 2: Numerator becomes 1 × √3 = √3

Step 3: Denominator becomes √3 × √3 = 3

Answer: √3/3

Grade 8

Expand and simplify (2 + √5)(3 − √5).

Step 1 — First: 2 × 3 = 6

Step 2 — Outer: 2 × (−√5) = −2√5

Step 3 — Inner: √5 × 3 = 3√5

Step 4 — Last: √5 × (−√5) = −5

Step 5 — Collect terms: 6 − 2√5 + 3√5 − 5 = 1 + √5

Answer: 1 + √5

Grade 9

Rationalise fully: 5/(3 + √2). Give your answer in the form a + b√2.

Step 1 — Identify conjugate: The conjugate of (3 + √2) is (3 − √2).

Step 2 — Multiply top and bottom: [5(3 − √2)] / [(3 + √2)(3 − √2)]

Step 3 — Expand numerator: 5 × 3 − 5 × √2 = 15 − 5√2

Step 4 — Expand denominator: 3² − (√2)² = 9 − 2 = 7

Step 5 — Write result: (15 − 5√2)/7

Answer: (15 − 5√2)/7 or equivalently 15/7 − (5/7)√2

Common Mistakes With Surds

Adding unlike surds: Writing √2 + √3 = √5 is wrong. You can only combine surds with the same radicand.

Not fully simplifying: Leaving √12 as 2√3 is correct, but leaving it as √12 loses the mark. Always look for perfect square factors.

Forgetting to multiply both top and bottom: When rationalising, you must multiply the numerator AND denominator by the same expression, or you change the value of the fraction.

Sign error in the conjugate: The conjugate of (a + √b) is (a − √b), not (−a + √b). Only the sign of the surd term changes.

Exam Tips

Memorise perfect squares up to 144: Quick recognition of 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144 speeds up surd simplification significantly.

Use FOIL for bracket expansion: Treat surd expansions exactly like algebraic bracket expansion. Write all four terms before collecting like terms.

Show every step when rationalising: Examiners award method marks for identifying the conjugate, expanding correctly, and simplifying. Do not skip steps.

Link surds to indices: Remember that √a = a^(1/2). This connection to fractional indices is often tested in Higher tier questions.