Surds

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Surds
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Simplifying a Surd
Rationalising a Surd
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Conjugate Pairs
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Starter Questions
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Use a calculator to find the values of
6
12
2
2
3
The Laws Of Surds
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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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What is a Surd
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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
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Adding Subtracting Surds
Note v2 v3 does not equal v5
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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.
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First Rule
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Examples
List the first 10 square numbers
1, 4, 9, 16, 25, 36, 49, 64, 81, 100
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Simplifying Square Roots
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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
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Have a go !
Think square numbers
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? 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
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What Goes In The Box ?
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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
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Starter Questions
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Simplify
2v5
3v2
¼
¼
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The Laws Of Surds
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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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12
Second Rule
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Examples
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Rationalising Surds
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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
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Rationalising Surds
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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
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Rationalising Surds
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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 )
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Rationalising Surds
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Lets try this one Remember multiply top and
bottom by root you are trying to remove
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Rationalising Surds
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Rationalise the denominator
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What Goes In The Box ?
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Rationalise the denominator of the following
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Starter Questions
Conjugate Pairs.
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Multiply out
3
14
12- 9 3
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The Laws Of Surds
Conjugate Pairs.
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Learning Intention
Success Criteria

1. To explain how to use the conjugate pair to
   rationalise a complex fractional surd.

• Know that
• (va vb)(va - vb) a - b

2. To be able to use the conjugate pair to
rationalise complex fractional surd.
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Looks something like the difference of two squares
Rationalising Surds
Conjugate Pairs.
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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 )
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Third Rule
Conjugate Pairs.
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Examples
7 3 4
11 5 6
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Rationalising Surds
Conjugate Pairs.
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Rationalise the denominator in the expressions
below by multiplying top and bottom by the
appropriate conjugate
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Rationalising Surds
Conjugate Pairs.
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Rationalise the denominator in the expressions
below by multiplying top and bottom by the
appropriate conjugate
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What Goes In The Box
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Rationalise the denominator in the expressions
below
Rationalise the numerator in the expressions
below