Where can I download free surds PDF worksheets with answers?

On this MathsLinks page you can download free printable surds PDF worksheets (Qs) and full worked solutions or mark schemes (Ans) from Ken Stafford, Maths Genie, Maths4Everyone, Corbett Practice, Corbett Textbook, plus a printable one-page surds revision guide.

What GCSE grade is surds (7A)?

Surds (topic code 7A) is a Grade 7 number topic in GCSE Maths for Edexcel, AQA, OCR, Eduqas and IGCSE specifications.

Free Surds Worksheets (PDF) with Answers — GCSE Maths

Free Surds PDF Worksheets & Answers

Grade 7 • Topic 7A

Resource	Questions	Answers
Ken Stafford GCSE past paper questions & solutions	Qs	Ans
Maths Genie Graded exam questions & solutions	Qs	Ans
Maths4Everyone GCSE 9–1 practice booklet & solutions	Qs	Ans
Corbett Practice Practice Questions & answers	Qs	Ans
Corbett Textbook Textbook Exercise & answers	Qs	Ans
Maths Made Easy MME worksheet & model answers	Qs	Ans
Dr Frost GCSE worksheet & solutions	Qs	Ans
1st Class Maths (Edxl) Edexcel exam questions & solutions	Qs	Ans
1st Class Maths (AQA) AQA exam questions & solutions	Qs	Ans
Metatutor Topic worksheet & worked solutions	Qs	Ans
Maths Takeaway GCSE Maths Takeaway exam questions	Qs	Ans
Dr Austin Practice grid / worksheet & answers	Qs	Ans
GCSE Maths Rocket Practice questions & answers	Qs	Ans
Maths DIY Exam-style booklet & solutions	Qs	Ans
IGCSE MT IGCSE (9–1) past paper booklet	Qs	Ans
E Classroom GCSE topic booklet & answers	Qs	Ans
PMT (Edxl) Edexcel GCSE topic questions & solutions	Qs	Ans
PMT (AQA) AQA GCSE topic questions & solutions	Qs	Ans
PMT (IGCSE) Edexcel IGCSE topic questions & solutions	Qs	Ans
The Maths Room Online topic notes & examples	Page
Cognito Interactive online lesson & practice	Page
esheets Self-marking online worksheet	Page

Grade 7 topics
7A - Surds	7B - Bounds
7C - Proportion (harder)	7D - Algebraic fractions
7E - The quadratic formula	7F - Factorising harder quadratics
7G - Trigonometric & exp. Graphs	7H - Rearranging (harder)
7J - Functions (harder)	7K - Iteration
7L - Area of any triangle	7M - The sine rule
7N - The cosine rule	7P - Congruent triangles
7Q - 3D pythagoras / trig.	7R - Histograms
7S - Conditional probability	7T - Exact trig. values (harder)
7U - Area of a segment

Explanation:

A surd is an irrational root that cannot be written as an exact fraction, such as √2, √3, or √5. To simplify a surd, split the number under the root into its largest square factor multiplied by another factor using √(ab) = √a × √b, and only collect like surds when adding or subtracting.

To rationalise the denominator of a fraction, remove the surd from the bottom: for a√b, multiply top and bottom by √b; for ca + √b, multiply top and bottom by the conjugate a − √b so the middle surd terms cancel out via the difference of two squares.

Example 1: Simplify fully √75 + √48 − 2√12

Solution:

Extract the largest square factor from each surd:

√75 = √25 × √3 = 5√3

√48 = √16 × √3 = 4√3

2√12 = 2 × √4 × √3 = 2 × 2√3 = 4√3

5√3 + 4√3 − 4√3 = 5√3

Answer: 5√3

Example 2: Rationalise the denominator of 143 − √2

Solution:

Multiply numerator and denominator by 3 + √2:

= 14(3 + √2)(3 − √2)(3 + √2)

Denominator: 3² − (√2)² = 9 − 2 = 7

= 14(3 + √2)7 = 2(3 + √2)

Answer: 6 + 2√2

Scanned GCSE Exam Questions & Worked Solutions

GCSE Past Paper Surds Questions & Answers

10-Question Interactive Quiz More Past Paper Questions

Past Paper Example 1 GCSE May 2024 1H Q17 3 marks

Full Paper (PDF) • Mark Scheme (PDF)

(a) Rationalise the denominator of 1/√7 (1) (b) Simplify fully √80 – √5 (2)

(a) Rationalise the denominator of 1√(7) [1 mark]

(b) Simplify fully √(80) - √(5) [2 marks]

Worked Solution & Mark Scheme (3 marks)

Part (a) Solution:

(1 × √7)/(√7 × √7) = √7/7 B1

Answer (a): √(7)7

Part (b) Solution:

√80 – √5 = √(16 × 5) – √5 = 4√5 – √5 = 3√5 M1 A1

Answer (b): 3√(5)

Past Paper Example 2 AQA Nov 2024 1H Q20 4 marks

Full Paper (PDF) • Mark Scheme (PDF)

(a) Show that √363 / √3 simplifies to an integer. [2 marks]
20 (b) Rationalise the denominator and simplify 20 / √5. [2 marks]

(a) Show that √(363)√(3) simplifies to an integer. [2 marks]

(b) Rationalise the denominator and simplify 20√(5) [2 marks]

Worked Solution & Mark Scheme (4 marks)

Part (a) Solution:

√363 / √3 = √(363 ÷ 3) = √121 = 11 M1 C1

Answer (a): √(363/3) = √121 = 11

Part (b) Solution:

(20 / √5) × (√5 / √5) = 20√5 / 5 = 4√5 M1 A1

Answer (b): Answer 4√(5)

Past Paper Example 3 AQA Jun 2023 1H Q22 3 marks

Full Paper (PDF) • Mark Scheme (PDF)

a = √(2) and b = √(18)
Match each expression to its value.
One has been done for you.

a = √2 and b = √18. Match each expression (a², a + b, ab, b/a) to its value. [3 marks]

Worked Solution & Mark Scheme (3 marks)

b = √18 = 3√2. M1
a + b = √2 + 3√2 = 4√2. M1
ab = √2 × √18 = √36 = 6.
b/a = √18 / √2 = √9 = 3. A1

Answer: a + b → 4√2, ab → 6, b/a → 3

← 6P Capture recapture

Back to GCSE questions by topic

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Surds

Free Surds PDF Worksheets & Answers

Guide to Surds: Explanation, Diagrams & Worked Examples

GCSE Past Paper Surds Questions & Answers