The Laws Of Surds.
3 5
11  2 3
5
7 3
9
21  7 3
11
7  10 13
3 5
What Is A Surd ?
Calculate the following roots:
(1) 36
=6
( 2) 8
( 3)4 81 ( 4)3 125
(5)5 32
=2
=3
=2
3
=5
All of the above roots have exact values and are called rational .
Now use a calculator to estimate the following roots:
(1) 2
( 2)3 21
( 3)4 100
(4) 83
 1.41
 2.76
 3.16
 9.11
All these roots do not have exact values and are called irrational .
They are called surds.
Adding & Subtracting Surds.
Because a surd such as 2 cannot be calculated exactly it can be
treated in the same way as an “x” variable in algebra. The following
examples will illustrate this point.
Simplify the following:
(1)4 2  6 2
( 2)16 23  7 23
 10 2
 9 23
Treat this expression the same as :
4 x + 6x = 10x
( 3)10 3  7 3  4 3
 13 3
Treat this expression the same as
16 x - 7x = 9x
(4)8 17  12 17
 4 17
Simplifying Square Roots.
Some square roots can be broken down into a mixture of integer
values and surds. The following examples will illustrate this idea:
Simplify:
(1) 12
To simplify 12 we must split 12 into factors
but at least one of the factors used must have
an exact square root.
= 4 x 3
Now simplify the square root.
= 2 3
From this example it can be appreciated that
you must use the square numbers as factors
in order to simplify the square root.
The square numbers are:
4,9,16,25,36,49,64,81,100,121,144,169,196,225…
(2)  45
(3)  32
(3)  72
= 9 x 5
= 16 x 2
= 4 x 18
= 35
= 42
= 2 x 9 x 2
= 2 x 3 x 2
(4)  2700
=  100 x 27
= 10 x  9 x  3
= 10 x 3 x 3
= 303
= 62
This example
demonstrates the
need to keep
looking for further
simplification.
Using 36 would
have saved time
What Goes In The Box ? 1
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
Rationalising Surds.
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 deno min ator .
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”.
To rationalise the denominator in the following multiply the top and
bottom of the fraction by the square root you are trying to remove:
3
(1)
5
3
5

x
5
5

3 5
5 5
3 5

5
5 x 5 =  25 = 5
( 2)

3
2 7
3 7
2 7 7
10
( 3)
7 5
10  5

7 5 5
3 7

2 7
10 5

75
3 7

14
2 5

7
What Goes In The Box ? 2.
Rationalise the denominator of the following expressions:
7
(1)
3
4
( 2)
6
7 3

3
2 6

3
( 4)
4
9 2
2 2

9
( 3)

14
3 10
7 10
15
2 5
( 5)
7 3
6 3
( 6)
11 2
2 15

21
3 6

11
Conjugate Pairs.
Consider the expression below:
( 3  6)( 3  6)
This is a conjugate pair. The brackets are identical apart from
the sign in each bracket .
Now observe what happens when the brackets are multiplied out:
( 3  6)( 3  6)
=
3 X 3 - 6 3 + 6 3 - 36
= 3 - 36
= -33
When the brackets are multiplied out the surds cancel out and we
end up seeing that the expression is rational . This result is used
throughout the following slide.
Rationalise the denominator in the expressions below by
multiplying top and bottom by the appropriate conjugate:
(1)
2
53
( 2)
2( 5  3)

( 5  3)( 5  3)

2( 5  3)

( 5  5  3 5  5  9)
2( 5  3)

( 5  9)

 ( 5  3)
2
7
( 3  2)
7( 3  2 )
( 3  2 )( 3  2 )
7( 3  2 )

( 3  2)
 7( 3  2 )
In both of the above examples the surds have
been removed from the denominator as required.
What Goes In The Box ? 3.
Rationalise the denominator in the expressions below :
5
(1)
( 7  2)
5( 7  2)

3
3
( 2)
( 3  2)
 3 6
7
( 3)
( 2 5  4)
7( 5  2 )

2
Rationalise the numerator in the expressions below :
( 4)

64
12
5
6( 6  4)
( 5)

5  11
7
6
7( 5  11 )
2 3  11
( 6)
3
1

6  33