Surds and Rationalising Denominators — Edexcel A-level Mathematics (Pure)

SPEC 2.2 • ALGEBRA AND FUNCTIONS

Surds and Rationalising the Denominator

Master exact values with surds and learn to rationalise denominators

What is a Surd?

Definition

A surd is an irrational root that cannot be simplified to a rational number. Surds are left in root form to give exact values.

Examples of surds:

2

,

3

,

5

,

37

,

11

NOT surds (they simplify to rational numbers):

4 = 2

,

9 = 3

,

25 = 5

Why Use Surds?

Surds give exact values rather than rounded decimals.

For example:

2 = 1.41421356...

(never-ending decimal) but

2

is exact.

Important

In exams, always leave answers as surds unless told to give a decimal. "Give your answer in the form

a + b c

" means use surds!

Key Surd Rules

These rules allow us to manipulate and simplify surds:

ab = a \times b

\frac{a}{b} = \frac{a}{b}

Like Surds

Surds with the same root can be added or subtracted:

p a + q a = (p + q) a

Key Skill

Simplifying surds: Find the largest square factor and extract it.

72 = 36 \times 2 = 36 \times 2 = 62

Common Error

a + b \neq = a + b

— surds do NOT distribute over addition!

Example:

9 + 16 = 25 = 5

, but

9 + 16 = 3 + 4 = 7

✗

Worked Example01

Simplify: (a) $72$ (b) $200$

(a) $72$

1

Find the largest square factor:

72 = 36 \times 2

2

Apply the product rule:

72 = 36 \times 2 = 36 \times 2

3

Simplify:

36 \times 2 = 62

(b) $200$

1

Find the largest square factor:

200 = 100 \times 2

2

Apply the product rule:

200 = 100 \times 2 = 100 \times 2

3

Simplify:

100 \times 2 = 102

Adding and Subtracting Surds

You can only add or subtract like surds (surds with the same root).

Rule

a n + b n = (a + b) n

a n - b n = (a - b) n

Method

Step 1: Simplify each surd first

Step 2: Combine like surds by adding/subtracting coefficients

Example

32 + 52 = 82

✓

32 + 53

cannot be simplified (different surds)

Worked Example02

Simplify: $312 + 227 - 48$

1

Simplify each surd:

•

12 = 4 \times 3 = 23

•

27 = 9 \times 3 = 33

•

48 = 16 \times 3 = 43

2

Substitute:

3 (23) + 2 (33) - (43)

3

Multiply coefficients:

63 + 63 - 43

4

Combine like surds:

(6 + 6 - 4) 3 = 83

Multiplying Surds

Use the product rule to multiply surds together:

Rules

a \times b = ab

a \times a = a

Expanding Brackets

When expanding brackets with surds, multiply each term as normal:

a (b + c) = b a + a c

Remember

(a)^{2} = a \times a = a

This is useful when simplifying expressions!

Worked Example03

Expand and simplify: $3 (2 + 12)$

1

Expand the bracket by multiplying each term by

3

:

•

3 \times 2 + 3 \times 12

2

Simplify the first term:

23

3

Simplify the second term:

3 \times 12 = 3 \times 12 = 36 = 6

4

Combine:

23 + 6

or

6 + 23

Worked Example04

Expand and simplify: $(3 + 5) (2 - 5)$

1

Use FOIL (First, Outer, Inner, Last):

•

First:

3 \times 2 = 6

•

Outer:

3 \times (- 5) = - 35

•

Inner:

5 \times 2 = 25

•

Last:

5 \times (- 5) = - (5)^{2} = - 5

2

Combine all terms:

6 - 35 + 25 - 5

3

Collect like terms:

(6 - 5) + (- 35 + 25) = 1 - 5

Rationalising the Denominator

Definition

Rationalising means removing the surd from the denominator of a fraction.

Why rationalise? It's considered "simplified form" and makes calculations easier.

Method for Single Surd

To rationalise

\frac{k}{a}

, multiply top and bottom by

a

:

\frac{k}{a} = \frac{k}{a} \times \frac{a}{a} = \frac{k a}{a}

Why It Works

Multiplying by

\frac{a}{a} = 1

doesn't change the value, but

a \times a = a

removes the surd from the denominator.

Worked Example05

Rationalise the denominator: $\frac{6}{3}$

1

Multiply top and bottom by

3

:

•

\frac{6}{3} \times \frac{3}{3}

2

Multiply numerator:

6 \times 3 = 63

3

Multiply denominator:

3 \times 3 = 3

4

Result:

\frac{6 3}{3}

5

Simplify:

\frac{6 3}{3} = 23

Worked Example06

Rationalise and express in the form $a 5$ : $\frac{10}{2 5}$

1

Multiply top and bottom by

5

:

•

\frac{10}{2 5} \times \frac{5}{5}

2

Multiply numerator:

10 \times 5 = 105

3

Multiply denominator:

25 \times 5 = 2 \times 5 = 10

4

Result:

\frac{10 5}{10}

5

Simplify:

\frac{10 5}{10} = 5

Conjugate Pairs

Definition

Conjugate pairs are expressions like

(a + b)

and

(a - b)

.

When we multiply conjugate pairs, the surds cancel out:

(a + b) (a - b) = a^{2} - b

This is the difference of two squares pattern.

For Rationalising

To rationalise

\frac{k}{a + b}

, multiply top and bottom by the conjugate

(a - b)

:

\frac{k}{a + b} \times \frac{a - b}{a - b} = \frac{k ( a - b )}{a ^{2} - b}

Worked Example07

Rationalise and express in the form $a + b 2$ : $\frac{3 + 2}{5 - 2}$

1

Multiply by the conjugate

\frac{5 + 2}{5 + 2}

:

•

\frac{3 + 2}{5 - 2} \times \frac{5 + 2}{5 + 2}

2

Denominator:

(5 - 2) (5 + 2) = 25 - 2 = 23

3

Numerator:

(3 + 2) (5 + 2)

•

= 15 + 32 + 52 + 2 = 17 + 82

4

Result:

\frac{17 + 8 2}{23}

5

In form

a + b 2

:

\frac{17}{23} + \frac{8}{23} 2

Worked Example08

Simplify: $\frac{1}{2 + 3} + \frac{1}{2 - 3}$

1

Rationalise each fraction separately:

•

\frac{1}{2 + 3} \times \frac{2 - 3}{2 - 3} = \frac{2 - 3}{4 - 3} = 2 - 3

•

\frac{1}{2 - 3} \times \frac{2 + 3}{2 + 3} = \frac{2 + 3}{4 - 3} = 2 + 3

2

Add the two rationalised fractions:

•

(2 - 3) + (2 + 3)

3

Simplify (surds cancel):

2 + 2 = 4

Worked Example09

Express in the form $a + b 5$ : $\frac{( 3 - 5 ) ^{2}}{2 + 5}$

1

Expand the numerator

(3 - 5)^{2}

:

•

= 9 - 65 + 5 = 14 - 65

2

Now rationalise

\frac{14 - 6 5}{2 + 5}

using conjugate

(2 - 5)

:

3

Denominator:

(2 + 5) (2 - 5) = 4 - 5 = - 1

4

Numerator:

(14 - 65) (2 - 5)

•

= 28 - 145 - 125 + 6 \times 5

•

= 28 - 265 + 30 = 58 - 265

5

Result:

\frac{58 - 26 5}{- 1} = - 58 + 265

Practice

Question 1: Simplify

125 + 320 - 45

1

Simplify each surd:

•

125 = 25 \times 5 = 55

•

20 = 4 \times 5 = 25

•

45 = 9 \times 5 = 35

2

Substitute:

55 + 3 (25) - 35

3

Simplify:

55 + 65 - 35 = 85

Practice

Question 2: Rationalise

\frac{15}{5}

and give your answer in its simplest form.

1

Multiply top and bottom by

5

:

•

\frac{15}{5} \times \frac{5}{5}

2

Numerator:

15 \times 5 = 155

3

Denominator:

5 \times 5 = 5

4

Result:

\frac{15 5}{5} = 35

Practice

Question 3: Rationalise

\frac{4}{3 - 7}

and express in the form

a + b 7

1

Multiply by the conjugate

\frac{3 + 7}{3 + 7}

:

•

\frac{4}{3 - 7} \times \frac{3 + 7}{3 + 7}

2

Denominator:

(3 - 7) (3 + 7) = 9 - 7 = 2

3

Numerator:

4 (3 + 7) = 12 + 47

4

Result:

\frac{12 + 4 7}{2} = 6 + 27

Summary

Key Rules

OperationRule

Product

ab = a \times b

Quotient

\frac{a}{b} = \frac{a}{b}

Like surds

a n \pm b n = (a \pm b) n

Rationalise (single)

\frac{k}{a} \to

multiply by

\frac{a}{a}

Rationalise (conjugate)

\frac{k}{a \pm b} \to

multiply by

\frac{a \mp b}{a \mp b}

Exam Tips

Always simplify surds first before adding/subtracting

Check your answer is in the required form (e.g.,

a + b c

)

Use conjugate pairs for denominators like

(a \pm b)

Remember:

a + b \neq = a + b