The Laws Of Surds.

1
The Laws Of Surds.
2
What Is A Surd ?
Calculate the following roots
3
5
2
6
2
All of the above roots have exact values and are
called rational .
Now use a calculator to estimate the following
roots
All these roots do not have exact values and are
called irrational .
They are called surds.
3
Adding Subtracting Surds.
Because a surd such as ?2 cannot be calculated
exactly it can be treated in the same way as an
x variable in algebra. The following examples
will illustrate this point.
Simplify the following
Treat this expression the same as
4 x 6x 10x
Treat this expression the same as
16 x - 7x 9x
4
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
Simplify
To simplify ?12 we must split 12 into factors
but at least one of the factors used must have an
exact square root.
(1) ?12
?4 x ?3
Now simplify the square root.
From this example it can be appreciated that you
must use the square numbers as factors in order
to simplify the square root.
2 ?3
The square numbers are
4,9,16,25,36,49,64,81,100,121,144,169,196,225
5
(3) ? 72
(2) ? 45
(3) ? 32
?4 x ?18
?9 x ?5
?16 x ?2
2 x ?9 x ?2
4?2
3?5
2 x 3 x ?2
(4) ? 2700
6?2
This example demonstrates the need to keep
looking for further simplification. Using ?36
would have saved time
? 100 x ?27
10 x ? 9 x ? 3
10 x 3 x ?3
30?3
6
What Goes In The Box ? 1
Simplify the following square roots
(2) ? 27
(3) ? 48
(1) ? 20
4?3
2?5
3?3
(6) ? 3200
(4) ? 75
(5) ? 4500
5?3
40?2
30?5
7
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.
Fractions can contain 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.
8
To rationalise the denominator in the following
multiply the top and bottom of the fraction by
the square root you are trying to remove
?5 x ?5 ? 25 5
9
What Goes In The Box ? 2.
Rationalise the denominator of the following
expressions
10
Conjugate Pairs.
Consider the expression below
This is a conjugate pair. The brackets are
identical apart from the sign in each bracket .
Now observe what happens when the brackets are
multiplied out

?3 X ?3
- 6 ?3
6 ?3
- 36
3 - 36
-33
When the brackets are multiplied out the surds
cancel out and we end up seeing that the
expression is rational . This result is used
throughout the following slide.
11
Rationalise the denominator in the expressions
below by multiplying top and bottom by the
appropriate conjugate
In both of the above examples the surds have been
removed from the denominator as required.
12
What Goes In The Box ? 3.
Rationalise the denominator in the expressions
below
Rationalise the numerator in the expressions
below