Surds
Wednesday, 11 June 2014
WALT:
To use the rules of surds and simplify a surd in
the form p q .
Definition of a surd
2 and 3 , etc are irrational numbers which are in surd form. They cannot
be written as integer values
Laws:
a  b  ab
a
a

a b 
b
b
The special case:
a a a
Examples: Write the following as simply as possible using the Laws
12 × 3 =
72
8
=
11 × 11 =
36 = 6
9
=3
121 = 11
4 2 × 3 18 = 4 × 3 × 36 = 72
Surds in the form p 𝑞
Using the known square numbers it is possible to
simply surd values in the form p 𝑞
Examples: Simplify the following in the form p 𝑞
12 = 4 × 3 = 2 3
Square root the square number.
Re-write 12 as a pair of multiples
containing the highest possible
square number.
45 = 9 × 5 = 3 5
Square root the square number.
Re-write 45 as a pair of multiples
containing the highest possible
square number.
242 = 121 × 2 = 11 2
Square root the square number.
Re-write 242 as a pair of
multiples containing the highest
possible square number.
72
82
92
112
= 49
= 64
= 81
= 121
700 = 100 × 7 = 10 7
Square root the square number.
Re-write 700 as a pair of multiples
containing the highest possible
square number.
180 = 36 × 5 = 6 5
Square root the square number.
Re-write 72 as a pair of multiples
containing the highest possible
square number.
72
82
92
112
= 49
= 64
= 81
= 121
Example
a) Simplify
b) Simplify
8
(i)
32 ii
Re-write 32 as a pair of multiples
containing the highest possible
square number.
50
(ii) Hence simplify 8 + 32 − 50
a) 8 = 4 × 2 = 2 2
Re-write 8 as a pair of multiples
containing the highest possible
square number.
b) 32= 16 × 2 = 4 2
50 = 25 × 2 = 5 2
Re-write 50 as a pair of multiples
containing the highest possible
square number.
8 + 32 − 50 = 2 2 + 4 2 − 5 2
= 2
Note: They should have
the same surd in each
term when simplified.
Example
a) Simplify
20
b) Hence simplify
(i)
20 + 45 − 125
(ii)
245 ÷ 20
a) 20 = 4 × 5 = 2 5
b) 45= 9 × 5 = 3 5
125= 25 × 5 = 5 5
20 + 45 − 125 = 2 5 + 3 5 − 5 5
=0
245 = 49 × 5 = 7 5
245 ÷ 20 = 7 5 ÷ 2 5
7
=
2
Rationalising the denominator
Wednesday, 11 June 2014
WALT:
To use the rules of surds to simplify brackets and
to rationalise a denominator.
Example
Rremove the brackets and simplify each of the
following
(i)
(ii)
(iii)
(iv)
(i)
 77 − 22 77 +22
 55+ 11 55+12
 3  2  3  2 
 3  1 3  4
7−2
7+2
𝟕
+2
𝟕
𝟕
+𝟐 𝟕
-2
−𝟐 𝟕
−𝟒
=3
𝟓
(ii)
5+1
5+2
=7+3 5
𝟓
+1
+2
𝟓 +𝟐 𝟓
+𝟏 𝟓 +𝟐
(iii)
3− 2
(iv)
3+1
3+ 2 =3− 6+ 6−2
3+4
=3+ 3+4 3+4
=1
=7+5 3
15
5
To make the denominator a
rational number, we can simply
multiply top and bottom by the
irrational number that is on the
denominator.
×
5
5
15 5
=
5
3 5
Simplify the numbers if possible
Example
Express with rational denominator
8
3
(i)
i)
8
3
8 3
×
=
3
3
3
44
(iii)
3 1212
16
(ii)
8
16 8
=
×
8
8
8
ii) 16
8
=2 8
=2 4×2
iii)
4
3 12
×
12
4 12
12
=
=
12
36
9
4×3
2 3
=
=
9
9
=4 2
Example
Simplify
12
75  27 
3
75  5 3
27  3 3
12
12 3
4 3


3
3
3
3
75  27 
12
3
5 3 3 3 4 3
4 3
Surds
Wednesday, 11 June 2014
WALT:
Rationalise a denominator with a compound surd
using the conjugate.
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 5 2)(
2 552)
2 = 5 2 5 2 5 4
= 5 4  1
When a conjugate pair is multiplied the surds ALWAYS
cancel out and we end up seeing that the expression
is rational ( no root sign ).
Multiply top and bottom by the
conjugate of the denominator
(i)
1
5 1
5 1
5 1



5 1
4
5 1
5 1

 

(ii)
4
7 3 4 7 3
4 7 3



73
4
7 3
7 3
(iii)
11  1 11  3 11  11  3 14  4 11
11  3



11  1
10
11  1
11  1
 7 3
Example
Rationalise the denominator and simplify each of the following:
(i)
(ii)
1
2−1
4
Multiply top and bottom by the
conjugate of the denominator
5+3
(i)
2 1



2 1
2 1
2 1
(ii)
4 5 3


59
5 3
5 3
1
4
2 1
5 3


2 1
 2 1
1


4 5 3

 3 5
4
Example
Rationalise the denominator and simplify each of the following:
(i)
(ii)
(i)
(ii)
7
2+ 3
2−3
2+1
7
2 3

2 3
2 3

14  7 3 14  7 3
 14  7 3

43
1
23 2  2 3


2 1
2 1
2 1
54 2

 54 2
1
2 3
2 1
A Level Past Paper Questions
1.
Simplify 4  3 , expressing your answer in surd form.
2.
Simplify the following
a)
35
b)
3 3
20 
7

20
5
2 3
52 3
3. Simplify each of the following expressions, expressing your answers in surd
form.
a) 2 32  3 8  18
b)
6  30
6  30