Surds – Grade 9 IGCSE Maths

Welcome to Surds!

A surd is a square root (or cube root, etc.) that cannot be simplified to a rational number. Surds arise naturally in geometry, trigonometry and algebra — they let us express exact answers without decimals. Mastering surds is a key IGCSE skill.

A surd is an irrational root | √2, √3, √5 are surds | √4 = 2 is NOT a surd

What is a Surd?

Rational vs irrational roots

Simplifying Surds

Find the largest square factor

Adding & Subtracting

Collect like surds

Multiplying Surds

√a × √b = √(ab)

Expanding Brackets

FOIL method with surds

Rationalising

Remove surds from denominators

1. What is a Surd?

A surd is a root of a positive integer that gives an irrational result — it cannot be expressed as an exact fraction or terminating decimal. The square root of any non-perfect-square integer is a surd.

NOT surds (rational): √1 = 1, √4 = 2, √9 = 3, √16 = 4, √25 = 5, √100 = 10
Surds (irrational): √2 ≈ 1.41421…, √3 ≈ 1.73205…, √5 ≈ 2.23606…, √7 ≈ 2.64575…

Quick test: is the number under the root a perfect square? If yes — not a surd. If no — it is a surd. Perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100…

2. Simplifying Surds

To write a surd in its simplest form, find the largest perfect square factor of the number under the root, then split the root using the multiplication rule.

√(a × b) = √a × √b ⟹ √n = k√m where k² is the largest square factor of n

Example: Simplify √72
Square factors of 72: 1, 4, 9, 36 ← largest is 36 = 6²
√72 = √(36 × 2) = √36 × √2 = 6√2

Example: Simplify √48
Square factors of 48: 1, 4, 16 ← largest is 16 = 4²
√48 = √(16 × 3) = 4√3

Always use the LARGEST square factor to get the simplest form in one step. Using a smaller factor works but takes more steps — e.g. √72 = √(4×18) = 2√18 = 2×3√2 = 6√2.

3. Adding and Subtracting Surds

Like terms in algebra can be collected. Surds behave the same way: you can only add or subtract surds with identical surd parts. Always simplify first.

a√n + b√n = (a + b)√n a√n − b√n = (a − b)√n

Example: √12 + √27
√12 = 2√3 √27 = 3√3
2√3 + 3√3 = 5√3

Example: √50 − √18
√50 = 5√2 √18 = 3√2
5√2 − 3√2 = 2√2

Cannot simplify: √2 + √3 — these are unlike surds, just as x + y cannot be simplified further.

4. Multiplying Surds

√a × √b = √(ab) and (√a)² = a and k√a × m√b = km√(ab)

√3 × √5 = √15
2√3 × 4√5 = 8√15
√6 × √6 = 6 (√a × √a = a always)
(3√2)² = 9 × 2 = 18

When multiplying surds, multiply the numbers outside the root together, and multiply the numbers inside the root together.

5. Expanding Brackets with Surds

Use the same FOIL technique as with algebraic expressions. Remember: (√a)² = a.

Example: (2 + √3)(1 + √3)
F: 2 × 1 = 2
O: 2 × √3 = 2√3
I: √3 × 1 = √3
L: √3 × √3 = 3
Total: 2 + 2√3 + √3 + 3 = 5 + 3√3

Special case — Difference of Two Squares:
(a + √b)(a − √b) = a² − (√b)² = a² − b (always rational!)
(3 + √5)(3 − √5) = 9 − 5 = 4

6. Rationalising the Denominator

A fraction is not in its simplest form if there is a surd in the denominator. Rationalising means rewriting the fraction so the denominator is rational (an integer).

Simple: multiply top and bottom by √a ⟹ 1/√a = √a/a

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

Conjugate: multiply by conjugate ⟹ 1/(a+√b) = (a−√b)/(a²−b)

Rationalise 1/(3 + √2):
Multiply by (3 − √2)/(3 − √2):
Numerator: 3 − √2
Denominator: (3 + √2)(3 − √2) = 9 − 2 = 7
Answer: (3 − √2)/7

The conjugate of (a + √b) is (a − √b). The product (a + √b)(a − √b) = a² − b, which is always rational.

Example 1 — Simplify √45

Square factors of 45: 1, 9. Largest = 9 = 3².

Split: √45 = √(9 × 5) = √9 × √5 = 3√5

Example 2 — Simplify √98

Square factors of 98: 1, 49. Largest = 49 = 7².

√98 = √(49 × 2) = 7√2. Answer: 7√2

Example 3 — Add: √12 + √27

√12 = 2√3 √27 = 3√3

2√3 + 3√3 = 5√3

Example 4 — Multiply: √2 × √8

√2 × √8 = √(2 × 8) = √16 = 4

Example 5 — Expand: (√5 + 1)(√5 − 1)

Difference of two squares: (√5)² − 1² = 5 − 1 = 4

Example 6 — Rationalise: 4/√2

Multiply by √2/√2: 4√2/(√2)² = 4√2/2 = 2√2

Example 7 — Rationalise: 1/(√3 + 1)

Conjugate = √3 − 1

Multiply: (√3 − 1)/((√3 + 1)(√3 − 1)) = (√3 − 1)/(3 − 1) = (√3 − 1)/2

Example 8 — Expand: (2 + √3)²

= 4 + 2·2·√3 + (√3)² = 4 + 4√3 + 3 = 7 + 4√3

Surd Simplifier — Step by Step

Enter any positive integer n to see a full step-by-step simplification of √n.

n =

Enter n and click Simplify to see all steps.

Surd Multiplication Tool

Multiply two surds and see the simplified result.

√ × √

Result will appear here.

Exercise 1 — Simplifying Surds

Write each surd in the form k√m. Enter the value of k (the integer coefficient).

1. Simplify √12. Write in the form k√3. Enter k.

2. Simplify √27. Write in the form k√3. Enter k.

3. Simplify √75. Write in the form k√3. Enter k.

4. Simplify √8. Write in the form k√2. Enter k.

5. Simplify √48. Write in the form k√3. Enter k.

6. Simplify √50. Write in the form k√2. Enter k.

7. Simplify √45. Write in the form k√5. Enter k.

8. Simplify √98. Write in the form k√2. Enter k.

9. Simplify √72. Write in the form k√2. Enter k.

10. Simplify √200. Write in the form k√2. Enter k.

Exercise 2 — Adding and Subtracting Surds

Simplify each expression fully. Give the coefficient of the simplified surd.

1. Simplify √12 + √27. Result = k√3. Find k.

2. Simplify √75 + √48. Result = k√3. Find k.

3. Simplify √12 − √3. Result = k√3. Find k.

4. Simplify √50 + √18. Result = k√2. Find k.

5. Simplify √8 + √2 + √18. Result = k√2. Find k.

6. Simplify 2√3 + √27. Result = k√3. Find k.

7. Simplify √50 − 3√2. Result = k√2. Find k.

8. Simplify √72 + √18. Result = k√2. Find k.

9. Simplify √75 + √48 + √3. Result = k√3. Find k.

10. Simplify 3√2 + √8. Result = k√2. Find k.

Exercise 3 — Multiplying Surds

Each product simplifies to an integer. Find that integer.

1. √2 × √8

2. √3 × √12

3. √5 × √20

4. √6 × √6

5. √7 × √7

6. 2√3 × √3

7. 3√3 × √3

8. √5 × √45

9. 2√2 × 2√2

10. √3 × √48

Exercise 4 — Rationalising the Denominator

Rationalise each denominator. Give the integer in the denominator after rationalising.

1. Rationalise 1/√2. Result = √2 / ? Enter the denominator.

2. Rationalise 1/√3. Result = √3 / ? Enter the denominator.

3. Rationalise 2/√5. Result = 2√5 / ? Enter the denominator.

4. Rationalise 3/√7. Result = 3√7 / ? Enter the denominator.

5. Rationalise 4/√2. Result = 4√2 / ? Enter denominator before simplifying.

6. Rationalise 6/√3. Result = 6√3 / ? Enter denominator before simplifying.

7. Rationalise 5/√5. Result = 5√5 / ? Enter denominator before simplifying.

8. Rationalise 6/√6. Result = 6√6 / ? Enter the denominator.

9. Rationalise 10/√10. Result = 10√10 / ? Enter the denominator.

10. Rationalise 11/√11. Result = 11√11 / ? Enter the denominator.

Exercise 5 — Expanding Brackets with Surds

Expand and simplify. Give the integer constant term of the result.

1. (1 + √3)(1 − √3). This is a² − b². Give the integer result.

2. (1 + √5)(1 − √5). Give the integer result.

3. (1 + √7)(1 − √7). Give the integer result.

4. (2 + √3)(2 − √3). Give the integer result.

5. (3 + √3)(3 − √3). Give the integer result.

6. (4 + √3)(4 − √3). Give the integer result.

7. (2 + √2)(2 − √2). Give the integer result.

8. (3 + √2)(3 − √2). Expand fully. Give integer constant term.

9. (√2 + 1)². Expand. The constant term is the integer part. Give it.

10. (√3 − 1)². Expand. Give the integer constant term.

Practice — 20 Questions

Mixed practice covering all surd skills. Read each question carefully.

1. Simplify √12. Give the coefficient k where √12 = k√3.

2. Simplify √27. Give the coefficient k.

3. Simplify √75. Give the coefficient k.

4. Simplify √8. Give the coefficient k where √8 = k√2.

5. Simplify √48. Give the coefficient k.

6. Simplify √50. Give the coefficient k where √50 = k√2.

7. Simplify √45. Give the coefficient k where √45 = k√5.

8. Simplify √98. Give the coefficient k.

9. Simplify √72. Give the coefficient k where √72 = k√2.

10. Simplify √200. Give the coefficient k.

11. √12 + √27. Give the coefficient of √3 in the answer.

12. √75 + √48. Give the coefficient of √3.

13. √12 − √3. Give the coefficient of √3.

14. √50 + √18. Give the coefficient of √2.

15. √8 + √2 + √18. Give the coefficient of √2.

16. 2√3 + √27. Give the coefficient of √3.

17. √50 − 3√2. Give the coefficient of √2.

18. √72 + √18. Give the coefficient of √2.

19. √75 + √48 + √3. Give the coefficient of √3.

20. 3√2 + √8. Give the coefficient of √2.

Challenge — 8 Questions

Harder surd problems. Each multiplication gives an integer — find it.

1. √2 × √8. Give the integer result.

2. √3 × √12. Give the integer result.

3. √5 × √20. Give the integer result.

4. √6 × √6. Give the integer result.

5. √7 × √7. Give the integer result.

6. 2√3 × √3. Give the integer result.

7. 3√3 × √3. Give the integer result.

8. √5 × √45. Give the integer result.