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SurdsSurds
Simplifying a Surd
Rationalising a Surd
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5. 2
Starter QuestionsStarter Questions
Use a calculator to find the values of :
1. 36 = 6
= 12
= 2
= 2
2. 144
33. 8 44. 16
1.41 2.7636. 21
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1. To explain what a surd is and to investigate the rules for surds.
1.1. Learn rules for surds.Learn rules for surds.
The Laws Of Surds
1.1. Use rules to simplify surds.Use rules to simplify surds.
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SurdsSurds
N = {natural numbers}
= {1, 2, 3, 4, ……….}
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
We can describe numbers by the following sets:
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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
SurdsSurdsS4 Credit
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SurdsSurds
QZWN
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, 35 etc
R
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2
What is a SurdWhat is a Surd
36 = 6
= 12
144
1.41 2.763 21
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
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
=10 2
16 23 - 7 23
=9 23
10 3 + 7 3 - 4 3 =13 3
Note :
√2 + √3 does not equal √5S4 Credit
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First Rule
4 6 24
a b ab
4 10 40
List the first 10 square numbers
Examples
1, 4, 9, 16, 25, 36, 49, 64, 81, 100
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Simplifying Square Roots
Some square roots can be broken down into a mixture of integer values and surds. The following examples will illustrate this idea:
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
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45 = 9 x 5= 35
32= 16 x 2= 42
72= 4 x 18
= 2 x 9 x 2= 2 x 3 x 2
= 62
Have a go !Think square numbersS4 Credit
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What Goes In The Box ?
Simplify the following square roots:
(1) 20 (2) 27 (3) 48
(4) 75 (5) 4500 (6) 3200
= 25
= 33
= 43
= 53
= 305 = 402
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First Rule
4 4 29 39
a abb
25 25 5 1100 10 2100
Examples
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Have a go !Think square numbersS4 Credit
481
4
81
29
818
2 4
2 9
23
10500
10
10 50
1
50
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Have a go !Think square numbersS4 Credit
2 15
3
2 3 5
3
2 5
5 5
2 20
5 5
2 4 5
54
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19 Apr 202319 Apr 2023 Created by Mr Lafferty Maths DeptCreated by Mr Lafferty Maths Dept
Now try MIA
Ex 7.1 Ex 8.1 Ch9 (page 185)
S4 Credit
Exact Values
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Starter QuestionsStarter Questions
Simplify :
1. 20 = 2√5
= 3√2
= ¼
2. 18
1 13.
2 2
1 14.
4 4 =
¼
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1. To explain how to rationalise a fractional surd.
1.1. Know that √a x √a = a.Know that √a x √a = a.
The Laws Of Surds
2.2. To be able to rationalise To be able to rationalise the numerator or the numerator or denominator of a denominator of a fractional surd.fractional surd.
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Second Rule
4 4 4
a a a
13 13 13
Examples
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Rationalising Surds
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 denominatorFractions can contain surds:
23
5
4 7
3 2
3 - 5
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Rationalising Surds
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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To rationalise the denominator multiply the top and bottom of the fraction by the square root you are
trying to remove:
3
53 5
=5 5
( 5 x 5 = 25 = 5 )
3 5=
5
Rationalising SurdsS4 Credit
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Let’s try this one :
Remember multiply top and bottom by root you are trying to remove
3
2 73 7
=2 7 7
3 7=
2 73 7
=14
Rationalising SurdsS4 Credit
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10
7 510 5
=7 5 5
10 5=
7 52 5
=7
Rationalising Surds
Rationalise the denominator
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What Goes In The Box ?
Rationalise the denominator of the following :
7
34
6
14
3 10
4
9 22 5
7 36 3
11 2
7 3=
32 6
=3
7 10=
15
2 29
2 15
=21
3 6=
11
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Conjugate Pairs.
Rationalising Surds
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 5 - 2 5 + 2 5 - 4
= 5 - 4
= 1When 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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Looks something like the difference of two squares
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Third Rule
7 3 7 3
a b a b a b
Examples
11 5 11 5
Conjugate Pairs.
= 7 – 3 = 4
= 11 – 5 = 6
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Rationalise the denominator in the expressions below by multiplying top and bottom by the
appropriate conjugate:
2
5 - 12( 5 + 1)
=( 5 - 1)( 5 + 1)
2( 5 + 1)=
( 5 5 - 5 + 5 - 1)2( 5 + 1)
=(5 - 1)
( 5 + 1)=
2
Conjugate Pairs.
Rationalising SurdsS4 Credit
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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)
Conjugate Pairs.
Rationalising SurdsS4 Credit
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What Goes In The Box
Rationalise the denominator in the expressions below :
5
( 7-2)3
( 3 - 2)
Rationalise the numerator in the expressions below :
6 + 412
5 + 117
= 3 + 6
- 5=6( 6 - 4)
- 6=7( 5 - 11)
5( 7 + 2)=
3
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19 Apr 202319 Apr 2023 Created by Mr Lafferty Maths DeptCreated by Mr Lafferty Maths Dept
Now try MIA
Ex 9.1 Ex 9.1 Ch9 (page 188)
S4 Credit
Rationalising Surds