IGCSE MATH: SURDS

Surds for IGCSE Maths (CIE)

What is a Surd?

A surd is an irrational number expressed as the root of a number that is not a perfect power. The most common surds are square roots of non-square numbers.

Examples of Surds: √2, √3, 5√7, ∛10
NOT Surds: √4 = 2, ∛27 = 3, √1 = 1 (These are rational numbers)

1. Simplifying Surds

The goal is to express a surd in its simplest form by identifying the largest square factor.

√a × b = √a × √b

Method:

Find the largest perfect square factor of the number under the root
Rewrite the surd as a product
Simplify the root of the perfect square

Example 1: Simplify √50

1√50 = √25 × 2

2= √25 × √2

3= 5 × √2

Answer: 5√2

Example 2: Simplify √72

1√72 = √36 × 2

2= √36 × √2

3= 6 × √2

Answer: 6√2

2. Manipulating Surds (The Four Operations)

A. Adding and Subtracting Surds

You can only combine surds that are "like terms" (they have the same irrational part).

a√b + c√b = (a + c)√b

Example 3: Simplify 3√5 + 7√5

3√5 + 7√5 = (3 + 7)√5 = 10√5

Example 4: Simplify 4√3 + 2√5 - √3 + 3√5

(4√3 - √3) + (2√5 + 3√5) = 3√3 + 5√5

This is fully simplified as √3 and √5 are different.

B. Multiplying Surds

Use the rule:

√a × √b = √a × b

Example 5: Simplify √3 × √12

√3 × √12 = √3 × 12 = √36 = 6

Example 6: Simplify (3√2) × (5√7)

(3 × 5) × (√2 × √7) = 15 × √14 = 15√14

Example 7: Expand and simplify √2(√8 - √3)

(√2 × √8) - (√2 × √3) = √16 - √6 = 4 - √6

C. Dividing Surds

Use the rule:

√a ÷ √b = √a ÷ b

Example 8: Simplify √20 ÷ √5

√20 ÷ √5 = √20 ÷ 5 = √4 = 2

3. Rationalising the Denominator

This is the process of removing a surd from the denominator of a fraction.

Case 1: Denominator is a Single Surd

Multiply the numerator and denominator by the surd in the denominator.

a √b

a × √b √b × √b

a√b b

Example 9: Rationalise

5 √3

√3 √3

5 × √3 √3 × √3

5√3 3

Case 2: Denominator is a Binomial

Multiply the numerator and denominator by the conjugate of the denominator.

Example 10: Rationalise

4 3 - √2

1Conjugate of (3 - √2) is (3 + √2)

2Multiply top and bottom:

4 3 - √2

3 + √2 3 + √2

3Numerator: 4 × (3 + √2) = 12 + 4√2

4Denominator: (3 - √2)(3 + √2) = 3² - (√2)² = 9 - 2 = 7

5Final Answer:

12 + 4√2 7

This can also be written as

12 7

4√2 7

4. Writing a Mixed Rational Number with a Surd

A mixed number containing a surd is an expression of the form a ± b√c, where a and b are rational numbers.

Example 11: Express

6 + 2√18 2

in the form a + b√c

1Simplify the surd first: √18 = √9 × 2 = 3√2

2Substitute:

6 + 2 × 3√2 2

6 + 6√2 2

3Divide both terms in the numerator by 2:

6 2

6√2 2

= 3 + 3√2

Final Answer: 3 + 3√2

Key Rules to Memorise

√a × √a = a
√a × √b = √ab
√a ÷ √b = √a/b
(a + b)(a - b) = a² - b² (The difference of two squares)

Exam Tips & Common Mistakes

Always Simplify First: Before any operation, check if the surds can be simplified. √12 + √27 = 2√3 + 3√3 = 5√3, not √39
Identify Like Terms: Only surds with the same number under the root can be added or subtracted
Rationalise Fully: A final answer must never have a surd in the denominator
Show Clear Working: When rationalising binomial denominators, writing the multiplication step clearly helps avoid errors

Summary of Process

Simplify the surd by taking out square factors
Identify like terms for addition/subtraction
Multiply/Divide using the rules √a × √b = √ab and √a ÷ √b = √a/b
Rationalise any fraction with a surd in the denominator
- Single surd denominator: Multiply by that surd
- Binomial denominator: Multiply by its conjugate

Good luck with your revision!