The Laws Of Surds
144 = 12
36 = 6
The above roots have exact values
and are called rational
2  1.41
3
21  2.76
These roots do NOT have exact values
and are called irrational OR
Surds
Adding and subtracting a surd such as 2. It can be
treated in the same way as an “x” variable in algebra.
The following examples will illustrate this point.
4 2+6 2
16 23 - 7 23
=10 2
10 3 + 7 3 - 4 3
=9 23
=13 3
a  b  ab
Examples
4  6  24
4  10  40
List the first 10 square numbers
1, 2, 4, 9, 16, 25, 36, 49, 64, 81, 100
Some square roots can be broken down into a
mixture of integer values and surds. The following
examples will illustrate this idea:
12
= 4 x 3
= 2 3
To simplify 12 we must split 12
into factors with at least one being
a square number.
Now simplify the square root.
Have a go !
Think square numbers
 45
 32
 72
= 9 x 5
= 16 x 2
= 4 x 18
= 35
= 42
= 2 x 9 x 2
= 2 x 3 x 2
= 62
Simplify the following square roots:
(1)  20
(2)  27
(3)  48
= 25
= 33
= 43
(4)  75
(5)  4500
(6)  3200
= 53
= 305
= 402
Simplify :
1.
3.
20 = 2√5 2.
1 1
 =
2 2
¼
18 = 3√2
1
1
4.

=
4
4
¼
a a  a
Examples
4 4  4
13  13  13
You may recall from your fraction work that the
top line of a fraction is the numerator and the
bottom line the denominator.
2
numerator
=
3 denominator
Fractions can contain surds:
2
3
5
4 7
3 2
3- 5
If by using certain maths techniques we remove the
surd from either the top or bottom of the fraction
then we say we are “rationalising the numerator” or
“rationalising the denominator”.
Remember the rule
a a  a
This will help us to rationalise a surd fraction
Rationalising Surds
To rationalise the denominator multiply the top and
bottom of the fraction by the square root you are
trying to remove:
3
3
5
=

5
5
5
3 5
=
5
( 5 x 5 =  25 = 5 )
Rationalising Surds
Let’s try this one :
Remember multiply top and bottom by root you are
trying to remove
3
3 7
3 7
3 7
=
=
=
14
2 7 2 7  7 2 7
Rationalising Surds
Rationalise the denominator
10
10  5
10 5 2 5
=
=
=
7 5 7 5  5 7 5
7
Rationalise the denominator of the following :
7
3
4
9 2
7 3
=
3
2 2

9
4
6
2 6
=
3
14
3 10
7 10
=
15
2 5
7 3
2 15
=
21
6 3
11 2
3 6
=
11
Conjugate Pairs.
Multiply out :
1.
3 3= 3
2.
14  14 = 14
3.

12 + 3


12 - 3 = 12- 9 = 3
Rationalising Surds
Conjugate Pairs.
Look at the expression :
( 5  2)( 5  2)
This is a conjugate pair. The brackets are identical
apart from the sign in each bracket .
Multiplying out the brackets we get :
( 5  2)( 5  2) = 5 x - 2 5 + 2 5 - 4
5
=5-4 =1
When the brackets are multiplied out the surds
ALWAYS cancel out and we end up seeing that the
expression is rational ( no root sign )
Conjugate Pairs.

a b
Examples



a  b  a b
7 3

7 3
=7–3=4

11  5


11  5
= 11 – 5 = 6

Rationalising Surds
Conjugate Pairs.
Rationalise the denominator in the expressions below by
multiplying top and bottom by the appropriate
conjugate:
2
5-1
2( 5 + 1)
=
( 5 - 1)( 5 + 1)
2( 5 + 1)
2( 5 + 1)
=
=
( 5  5 - 5 + 5 - 1)
(5 - 1)
( 5 + 1)
=
2
Rationalising Surds
Conjugate Pairs.
Rationalise the denominator in the expressions below
by multiplying top and bottom by the appropriate
conjugate:
7
( 3 - 2)
7( 3 + 2)
=
( 3 - 2)( 3 + 2)
7( 3 + 2)
=
(3 - 2)
= 7( 3 + 2)
Rationalise the denominator in the expressions below :
5
( 7-2)
5( 7 + 2)
=
3
3
=3+
( 3 - 2)
6
Rationalise the numerator in the expressions below :
6+4
12
-5
=
6( 6 - 4)
5 + 11
7
-6
=
7( 5 - 11)