The Laws Of Surds.
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Transcript The Laws Of Surds.
Surds
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S5 Int2
Simplifying a Surd
Rationalising a Surd
Conjugate Pairs
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Starter Questions
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S5 Int2
Use a calculator to find the values of :
1.
3.
5.
3
36 = 6
2.
8
144 = 12
4.
4
16
2 1.41 6.
3
21 2.76
=2
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=2
The Laws Of Surds
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S5 Int2
Learning Intention
Success Criteria
1. To explain what a surd is
and to investigate the
rules for surds.
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1. Learn rules for surds.
1. Use rules to simplify surds.
What is a Surd
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144 = 12
36 = 6
The above roots have exact values
and are called rational
2 1.41
3
21 2.76
These roots do NOT have exact values
and are called irrational OR
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Surds
Note :
Adding & Subtracting
√2 +Surds
√3 does not
equal √5
S5 Int2
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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.
4 2+6 2
16 23 - 7 23
=10 2
=9 23
10 3 + 7 3 - 4 3
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=13 3
First Rule
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a b ab
Examples
4 10 40
4 6 24
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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S5 Int2
Some square roots can be broken down into a
mixture of integer values and surds. The following
examples will illustrate this idea:
12
= 4 x 3
= 2 3
To simplify 12 we must split 12
into factors with at least one being
a square number.
Now simplify the square root.
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Have a go !
Think square numbers
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S5 Int2
45
32
72
= 9 x 5
= 16 x 2
= 4 x 18
= 35
= 42
= 2 x 9 x 2
= 2 x 3 x 2
= 62
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What Goes In The Box ?
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S5 Int2
Simplify the following square roots:
(1) 20
(2) 27
(3) 48
= 25
= 33
= 43
(4) 75
(5) 4500
(6) 3200
= 53
= 305
= 402
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Starter Questions
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S5 Int2
Simplify :
1.
3.
20 = 2√5 2.
1 1
=
2 2
¼
18 = 3√2
1
1
4.
=
4
4
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¼
The Laws Of Surds
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S5 Int2
Learning Intention
Success Criteria
1. To explain how to
rationalise a fractional
surd.
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1. Know that √a x √a = a.
2. To be able to rationalise the
numerator or denominator of
a fractional surd.
Second Rule
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S5 Int2
a a a
Examples
13 13 13
4 4 4
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Rationalising Surds
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S5 Int2
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 denominator
Fractions can contain surds:
2
3
5
4 7
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3 2
3- 5
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
a a a
This will help us to rationalise a surd fraction
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Rationalising Surds
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S5 Int2
To rationalise the denominator multiply the top and
bottom of the fraction by the square root you are
trying to remove:
3
3
5
=
5
5
5
3 5
=
5
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( 5 x 5 = 25 = 5 )
Rationalising Surds
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S5 Int2
Let’s try this one :
Remember multiply top and bottom by root you are
trying to remove
3
3 7
3 7
3 7
=
=
=
14
2 7 2 7 7 2 7
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Rationalising Surds
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Rationalise the denominator
10
10 5
10 5 2 5
=
=
=
7 5 7 5 5 7 5
7
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What Goes In The Box ?
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S5 Int2
Rationalise the denominator of the following :
7
3
4
9 2
7 3
=
3
2 2
9
4
6
2 6
=
3
14
3 10
7 10
=
15
2 5
7 3
2 15
=
21
6 3
11 2
3 6
=
11
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Starter Questions
Conjugate Pairs.
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S5 Int2
Multiply out :
1.
3 3= 3
2.
14 14 = 14
3.
12 + 3
12 - 3 = 12- 9 = 3
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The Laws Of Surds
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S5 Int2
Conjugate Pairs.
Learning Intention
Success Criteria
1. To explain how to use the
conjugate pair to
rationalise a complex
fractional surd.
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1. Know that
(√a + √b)(√a - √b) = a - b
2. To be able to use the
conjugate pair to rationalise
complex fractional surd.
Looks
something like
the difference
S5 Int2of two squares
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Rationalising Surds
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 2)( 5 2) = 5 x - 2 5 + 2 5 - 4
5
=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
a b
a b a b
7 3
7 3
11 5
11 5
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=7–3=4
= 11 – 5 = 6
Rationalising Surds
Conjugate Pairs.
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S5 Int2
Rationalise the denominator in the expressions below by
multiplying top and bottom by the appropriate
conjugate:
2
5-1
2( 5 + 1)
=
( 5 - 1)( 5 + 1)
2( 5 + 1)
2( 5 + 1)
=
=
( 5 5 - 5 + 5 - 1)
(5 - 1)
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( 5 + 1)
=
2
Rationalising Surds
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S5 Int2
Conjugate Pairs.
Rationalise the denominator in the expressions below
by multiplying top and bottom by the appropriate
conjugate:
7
( 3 - 2)
7( 3 + 2)
=
( 3 - 2)( 3 + 2)
7( 3 + 2)
=
(3 - 2)
= 7( 3 + 2)
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What Goes In The Box
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S5 Int2
Rationalise the denominator in the expressions below :
5
( 7-2)
5( 7 + 2)
=
3
3
=3+
( 3 - 2)
6
Rationalise the numerator in the expressions below :
6+4
12
-5
=
6( 6 - 4)
5 + 11
7
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-6
=
7( 5 - 11)