Transcript Document

Week 9 - Surds
16 July 2015
16/07/2015
1
Contents
Simplifying a Surd
 Rationalising a Surd
 Conjugate Pairs
 Trial & Improvement

16/07/2015
2
Starter Questions

Use a calculator to find the values of :
3
36
=6
144
8
=3
4
2  1.41
3
16
= 12
=2
21  2.76
What is a Surd ?

These roots have exact values and are
called rational
144 = 12
36 = 6

These roots do NOT have exact values
and are called irrational OR Surds
2  1.41
3
21  2.76
Adding & Subtracting Surds
Note :
√2 + √3 does not
equal √5
To add or subtract surds such as 2,
treat as a single object.
 Eg. 4 2  6 2
16 23  7 23

 10 2
10 3  7 3  4 3
 9 23
 13 3
Multiplying Surds
a  b  ab
•Eg
4  6  24
4  10  40
•List the first 10 square numbers
•1, 2, 4, 9, 16, 25, 36, 49, 64, 81, 100
Simplifying Surds

Some square roots can be simplified
by using this rule -
12
To simplify 12 we must split 12 into
factors with at least one being a square
number.
= 4 x 3
Now simplify the square root.
= 2 3
Have a go 
You need to look for 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
Simplifying Surds

Simplify the following square roots :

(1)  20
(2)  27
= 25
= 33
(4)  75
(5)  4500
= 53
= 305

(3)  48
= 43
(6)  3200
= 402
Starter Questions

Simplify :
√20
1 x 1
2
2
= 2√5
=¼
√18 = 3√2
1 x 1 =
√4 √4
¼
Second Rule
a a  a
Examples
4 4  4
13  13  13
Rationalising Surds

Remember fractions –
1
2

Numerator
Denominator
Fractions can contain surds in the
numerator, denominator or both:
3
5
5
4 3
3 2
3 5
Rationalising Surds

Removing the surd form numerator or
denominator
a  b  ab

Remember the rules

This will help us to rationalise a surd
fraction
a a  a
Rationalising Surds

Multiply top and bottom by the square
root you are trying to remove:
3
5
3

5
Multiply top and bottom by √5
5
5

3 5
5
Remember
5 x 5 =  25 = 5 )
Rationalising Surds

Remember multiply top and bottom by
root you are trying to remove
3
2 7

3
2 7

7

7
3 7
2 7

3 7
14
Rationalising Surds

Rationalise the denominator
10
7 5

10

7 5
5
5
10 5
7 5

2 5
7
Rationalise the Denominator
7
3
4
9 2
7 3
=
3
2 2

9
4
6
2 5
7 3
2 6
=
3
2 15
=
21
14
3 10
6 3
11 2
7 10
=
15
=
3 6
11
Conjugate Pairs - Starter Questions

Multiply out :
3 3
14 14
=3
= 14
( 12  3)( 12  3)
Conjugate Pairs.
This is a conjugate pair. (5 + 2)(5 - 2)
 The brackets are identical apart from the
sign in each bracket .
 Multiplying out the brackets we get :

5 x 5 - 2 5 + 2 5 - 4

=5-4 =1
When the brackets are multiplied out the
surds ALWAYS cancel out leaving a
rational expression
Conjugate Pairs - Third Rule


a b

a  b  a b


=7–3=4
Eg.

7 3


11  5

7 3
11  5

= 11 – 5 = 6
Rationalising Surds

Rationalise the denominator in the
expressions below by multiplying top and
bottom by the appropriate conjugate:
2
5 1


2 ( 5 1)
51
2
5 1


5 1
5 1
2 ( 5 1)
4


51
2
Rationalising Surds

Another one ...
7
( 3  2)


7 ( 3  2)
( 3 2 )
7
( 3  2)

( 3  2)
( 3

2)
 7 ( 3  2)
Rationalising the Denominator
Rationalise the denominator in the
expressions below :
3
5( 7 + 2)
5
=3+
=
3
( 3 2 )
( 7 2)

6 4
-5
=
12
6( 6 - 4)
5 11
7
6
-6
=
7( 5 - 11)
Trial and Improvement
A method which involves making a
guess and then systematically improving
it until you reach the answer
 Eg.
x 2 + 5 = 24
What is x?
 Make an initial guess, maybe x = 3
 Try it and then keep improving the guess

16/07/2015
24
Trial and Improvement
Try
Working Out
x2 + 5
Result
x=3
32 + 5 = 14
Too small
x=4
42 + 5 = 21
Too small
x=5
52 + 5 = 30
Too big
x = 4.5
4.52 + 5 = 25.25
Too big
x = 4.4
4.42 + 5 = 24.36
Too big
x = 4.3
4.32 + 5 = 23.49
Too small
16/07/2015
25
Trial and Improvement

There is an answer between 4.3 and 4.4
x = 4.35
4.352 + 5 = 23.9225
Too small
x = 4.36
4.362 + 5 = 24.0096
Too big

So x= 4.36 to 2 dp
16/07/2015
26
Session Summary
Surds
 Simplifying Surds
 Rationalising Surds
 Conjugate Pairs
 Trail & Improvement

16/07/2015
27