Surds

1
Surds
S4 Credit
Simplifying a Surd
Rationalising a Surd
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2
Starter Questions
S4 Credit
Use a calculator to find the values of
6
12
2
2
3
The Laws Of Surds
S4 Credit
Learning Intention
Success Criteria

1. To explain what a surd is and to investigate the
   rules for surds.

1. Learn rules for surds.

1. Use rules to simplify surds.

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4
Surds
S4 Credit
We can describe numbers by the following sets
1, 2, 3, 4, .
N natural numbers
W whole numbers
0, 1, 2, 3, ..
Z integers
.-2, -1, 0, 1, 2, ..
Q rational numbers
This is the set of all numbers which can be
written as fractions or ratios. eg 5 5/1
-7 -7/1 0.6 6/10 3/5
55 55/100 11/20 etc
5
Surds
S4 Credit
R real numbers
This is all possible numbers. If we plotted
values on a number line then each of the previous
sets would leave gaps but the set of real numbers
would give us a solid line.
We should also note that
N fits inside W W fits inside Z
Z fits inside Q Q fits inside R
6
Surds
Q
Z
W
N
R
When one set can fit inside another we say that
it is a subset of the other.
The members of R which are not inside Q are
called irrational (Surd) numbers. These cannot
be expressed as fractions and include ? ,?2,
3?5 etc
7
What is a Surd
S4 Credit
12
6
The above roots have exact values and are called
rational
These roots do NOT have exact values and are
called irrational OR
Surds
8
Adding Subtracting Surds
Note v2 v3 does not equal v5
S4 Credit
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.
9
First Rule
S4 Credit
Examples
List the first 10 square numbers
1, 4, 9, 16, 25, 36, 49, 64, 81, 100
10
Simplifying Square Roots
S4 Credit
Some square roots can be broken down into a
mixture of integer values and surds. The
following examples will illustrate this idea
To simplify ?12 we must split 12 into factors
with at least one being a square number.
?12
?4 x ?3
Now simplify the square root.
2 ?3
11
Have a go !
Think square numbers
S4 Credit
? 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
12
What Goes In The Box ?
S4 Credit
Simplify the following square roots
(2) ? 27
(3) ? 48
(1) ? 20
2?5
3?3
4?3
(6) ? 3200
(4) ? 75
(5) ? 4500
30?5
40?2
5?3
13
First Rule
S4 Credit
Examples
14
Have a go !
Think square numbers
S4 Credit
15
Have a go !
Think square numbers
S4 Credit
16
Exact Values
S4 Credit
Now try MIA Ex 7.1 Ex 8.1 Ch9 (page 185)
17
Starter Questions
S4 Credit
Simplify
2v5
3v2
¼
¼
18
The Laws Of Surds
S4 Credit
Learning Intention
Success Criteria

1. To explain how to rationalise a fractional surd.

1. Know that va x va a.

2. To be able to rationalise the numerator or
denominator of a fractional surd.
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19
Second Rule
S4 Credit
Examples
20
Rationalising Surds
S4 Credit
You may recall from your fraction work that the
top line of a fraction is the numerator and the
bottom line the denominator.
Fractions can contain surds
21
Rationalising Surds
S4 Credit
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
This will help us to rationalise a surd fraction
22
Rationalising Surds
S4 Credit
To rationalise the denominator multiply the top
and bottom of the fraction by the square root you
are trying to remove
( ?5 x ?5 ? 25 5 )
23
Rationalising Surds
S4 Credit
Lets try this one Remember multiply top and
bottom by root you are trying to remove
24
Rationalising Surds
S4 Credit
Rationalise the denominator
25
What Goes In The Box ?
S4 Credit
Rationalise the denominator of the following
26
Looks something like the difference of two squares
Rationalising Surds
Conjugate Pairs.
S4 Credit
Look at the expression
This is a conjugate pair. 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 and we end up seeing that the
expression is rational ( no root sign )
27
Third Rule
Conjugate Pairs.
S4 Credit
Examples
7 3 4
11 5 6
28
Rationalising Surds
Conjugate Pairs.
S4 Credit
Rationalise the denominator in the expressions
below by multiplying top and bottom by the
appropriate conjugate
29
Rationalising Surds
Conjugate Pairs.
S4 Credit
Rationalise the denominator in the expressions
below by multiplying top and bottom by the
appropriate conjugate
30
What Goes In The Box
S4 Credit
Rationalise the denominator in the expressions
below
Rationalise the numerator in the expressions
below
31
Rationalising Surds
S4 Credit
Now try MIA Ex 9.1 Ex 9.1 Ch9 (page 188)