Transcript The Laws Of Surds.

Surds
www.mathsrevision.com
S4 Credit
Simplifying a Surd
Rationalising a Surd
www.mathsrevision.com
Starter Questions
www.mathsrevision.com
S4 Credit
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
www.mathsrevision.com
=2
The Laws Of Surds
www.mathsrevision.com
S4 Credit
Learning Intention
Success Criteria
1. To explain what a surd is
and to investigate the
rules for surds.
www.mathsrevision.com
1. Learn rules for surds.
1. Use rules to simplify surds.
Surds
S4 Credit
www.mathsrevision.com
We can describe numbers by the following sets:
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 =
55% =
-7/
1
55/
100
0.6 = 6/10 = 3/5
=
11/
20
etc
Surds
www.mathsrevision.com
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
www.mathsrevision.com
Surds
N
W
Z
Q
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
What is a Surd
www.mathsrevision.com
S4 Credit
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
www.mathsrevision.com
Surds
Note :
Adding & Subtracting
√2 +Surds
√3 does not
equal √5
S4 Credit
www.mathsrevision.com
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
www.mathsrevision.com
=13 3
First Rule
www.mathsrevision.com
S4 Credit
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
www.mathsrevision.com
Simplifying Square Roots
www.mathsrevision.com
S4 Credit
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.
www.mathsrevision.com
Have a go !
Think square numbers
www.mathsrevision.com
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
www.mathsrevision.com
What Goes In The Box ?
www.mathsrevision.com
S4 Credit
Simplify the following square roots:
(1)  20
(2)  27
(3)  48
= 25
= 33
= 43
(4)  75
(5)  4500
(6)  3200
= 53
= 305
= 402
www.mathsrevision.com
First Rule
www.mathsrevision.com
S4 Credit
Examples
4
4 2


9
9 3
a
a

b
b
25
25 5 1

 
100
100 10 2
www.mathsrevision.com
Have a go !
Think square numbers
www.mathsrevision.com
S4 Credit
4
81
8
18
4

81
2 4

2 9
2

9
2

3
10
500
10

10  50
1

50
Have a go !
www.mathsrevision.com
S4 Credit
Think square numbers
2 15
3
5 5
2 20
2 3 5

3
5 5

2 4  5
2 5
5

4
Exact Values
S4 Credit
www.mathsrevision.com
Now try MIA
Ex 7.1 Ex 8.1
Ch9 (page 185)
21-Jul-15
Created by Mr Lafferty Maths Dept
Starter Questions
www.mathsrevision.com
S4 Credit
Simplify :
1.
3.
20 = 2√5 2.
1 1
 =
2 2
¼
18 = 3√2
1
1
4.

=
4
4
www.mathsrevision.com
¼
The Laws Of Surds
www.mathsrevision.com
S4 Credit
Learning Intention
Success Criteria
1. To explain how to
rationalise a fractional
surd.
www.mathsrevision.com
1. Know that √a x √a = a.
2. To be able to rationalise the
numerator or denominator of
a fractional surd.
Second Rule
www.mathsrevision.com
S4 Credit
a a  a
Examples
13  13  13
4 4  4
www.mathsrevision.com
Rationalising Surds
www.mathsrevision.com
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.
2
numerator
=
3 denominator
Fractions can contain surds:
2
3
5
4 7
www.mathsrevision.com
3 2
3- 5
Rationalising Surds
www.mathsrevision.com
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
a a  a
This will help us to rationalise a surd fraction
www.mathsrevision.com
Rationalising Surds
www.mathsrevision.com
S4 Credit
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
www.mathsrevision.com
( 5 x 5 =  25 = 5 )
Rationalising Surds
www.mathsrevision.com
S4 Credit
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
www.mathsrevision.com
Rationalising Surds
www.mathsrevision.com
S4 Credit
Rationalise the denominator
10
10  5
10 5 2 5
=
=
=
7 5 7 5  5 7 5
7
www.mathsrevision.com
What Goes In The Box ?
www.mathsrevision.com
S4 Credit
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
www.mathsrevision.com
Looks
something like
the difference
S4 Credit
of two squares
www.mathsrevision.com
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 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 )
www.mathsrevision.com
Third Rule
Conjugate Pairs.
www.mathsrevision.com
S4 Credit

Examples
a b


a  b  a b

7 3

7 3

11  5

11  5
www.mathsrevision.com

=7–3=4

= 11 – 5 = 6
Rationalising Surds
Conjugate Pairs.
www.mathsrevision.com
S4 Credit
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)
www.mathsrevision.com
( 5 + 1)
=
2
Rationalising Surds
www.mathsrevision.com
S4 Credit
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)
www.mathsrevision.com
What Goes In The Box
www.mathsrevision.com
S4 Credit
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
www.mathsrevision.com
-6
=
7( 5 - 11)
Rationalising Surds
S4 Credit
www.mathsrevision.com
Now try MIA
Ex 9.1 Ex 9.1
Ch9 (page 188)
21-Jul-15
Created by Mr Lafferty Maths Dept