Surds and rationalising the denominator

A LEVEL LINKS
Scheme of work: 1a. Algebraic expressions – basic algebraic manipulation, indices and surds

Key points
• A surd is the square root of a number that is not a square number,
for example 2, 3, 5, etc.
• Surds can be used to give the exact value for an answer.
• ab = a × b
a a
• =
b b
• To rationalise the denominator means to remove the surd from the denominator of a fraction.
a
• To rationalise you multiply the numerator and denominator by the surd b
b
a
• To rationalise you multiply the numerator and denominator by b − c
b+ c

Examples
Example 1 Simplify 50

50 = 25 × 2 1 Choose two numbers that are

factors of 50. One of the factors
must be a square number
= 25 × 2 2 Use the rule ab = a × b
= 5× 2 3 Use 25 = 5
=5 2

Example 2 Simplify 147 − 2 12

147 − 2 12 1 Simplify 147 and 2 12 . Choose

two numbers that are factors of 147
= 49 × 3 − 2 4 × 3
and two numbers that are factors of
12. One of each pair of factors must
be a square number

= 49 × 3 − 2 4 × 3 2 Use the rule ab = a × b

= 7× 3 − 2× 2× 3 3 Use 49 = 7 and 4 = 2
=7 3 −4 3
4 Collect like terms
=3 3
Example 3 Simplify ( 7+ 2 )( 7− 2 )
1 Expand the brackets. A common
( 7+ 2 )( 7− 2 ) 2

= 49 − 7 2 + 2 7 − 4
mistake here is to write ( 7) = 49

=7–2 2 Collect like terms:

=5 − 7 2+ 2 7
=− 7 2 + 7 2 =0

1
Example 4 Rationalise
3

1 1 3 1 Multiply the numerator and

= ×
3 3 3 denominator by 3

1× 3
= 2 Use 9 =3
9

3
=
3

2
Example 5 Rationalise and simplify
12

2 2 12 1 Multiply the numerator and

= ×
12 12 12 denominator by 12

2 × 4×3 2 Simplify 12 in the numerator.

=
12 Choose two numbers that are factors
of 12. One of the factors must be a
square number

3 Use the rule ab = a × b

2 2 3
= 4 Use 4 =2
12
5 Simplify the fraction:
2 3 2 1
= simplifies to
6 12 6
 3
Example 6 Rationalise and simplify
2+ 5

3 3 2− 5 1 Multiply the numerator and

= ×
2+ 5 2+ 5 2− 5 denominator by 2− 5

=
(
3 2− 5 )
( 2 + 5 )( 2 − 5 ) 2 Expand the brackets

6−3 5
=
4+ 2 5 −2 5 −5 3 Simplify the fraction

6−3 5
=
−1 4 Divide the numerator by −1
Remember to change the sign of all
= 3 5 −6 terms when dividing by −1

Practice
1 Simplify. Hint
a 45 b 125 One of the two
c 48 d 175 numbers you
choose at the start
e 300 f 28 must be a square
g 72 h 162 number.

2 Simplify. Watch out!

a 72 + 162 b 45 − 2 5 Check you have
chosen the highest
c 50 − 8 d 75 − 48
square number at
e 2 28 + 28 f 2 12 − 12 + 27 the start.

3 Expand and simplify.

a ( 2 + 3)( 2 − 3) b (3 + 3)(5 − 12)
c (4 − 5)( 45 + 2) d (5 + 2)(6 − 8)
4 Rationalise and simplify, if possible.
1 1
a b
5 11
2 2
c d
7 8
2 5
e f
2 5
8 5
g h
24 45

5 Rationalise and simplify.

1 2 6
a b c
3− 5 4+ 3 5− 2

Extend

6 Expand and simplify ( x+ y )( x− y )

7 Rationalise and simplify, if possible.
1 1
a b
9− 8 x− y
Answers

1 a 3 5 b 5 5
c 4 3 d 5 7
e 10 3 f 2 7
g 6 2 h 9 2

2 a 15 2 b 5
c 3 2 d 3
e 6 7 f 5 3

3 a −1 b 9− 3
c 10 5 − 7 d 26 − 4 2

5 11
4 a b
5 11
2 7 2
c d
7 2
e 2 f 5
3 1
g h
3 3

3+ 5 2(4 − 3) 6(5 + 2)
5 a b c
4 13 23

6 x−y

x+ y
7 a 3+ 2 2 b
x− y