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Higher Mathematics
Surds
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Surds
Higher Mathematics
What are Surds
A surd is a square root
which cannot be evaluated
without approximation.
Surds
Higher Mathematics
What are Surds
A surd is an irrational number
An irrational number cannot be written
as a simple fraction i.e. as
a
b
It is a non-recurring decimal
Surds
Higher Mathematics
Why are surds of interest
A surd is a square root
which cannot be evaluated without approximation.
These often occur when using
Pythagoras’ Theorem
Trigonometry
Using surds allows us to be EXACT
Surds
Higher Mathematics
Examples of Surds
2
3
5
6
etc.
A good rule of thumb is to think of a surd as:
Any root that cannot be evaluated as a whole number.
For example:
4
is not a surd
Surds
Higher Mathematics
Which of these are surds ?
7
20
Yes
Yes
25
No
32
81
Yes
No
Surds
Higher Mathematics
Rules of surds
We can add or subtract surds
if they are the same.
7 7 
2 7
7 54 5 
3 5
Just like algebra:
3x  2 x  5 x
Surds
Higher Mathematics
Examples
Simplify
35 32 3
7 5 2 2 3 5
Surds
Higher Mathematics
Rules of surds
We can multiply surds
5 3 
15
Let’s check this out with simple numbers
9 4 
3 
2 
36
6
Surds
Higher Mathematics
Rules of surds
We can multiply surds
in general
a b 
ab
Note that this works both ways
ab  a  b
Surds
Higher Mathematics
WARNING
a b 
a b
Let’s check this out with simple numbers
9 4 
13
3  2  5 
13
Surds
Higher Mathematics
Examples
Simplify
7 
3
8 
4
5 
7
Surds
Higher Mathematics
Rules of surds
We can divide surds
5

2
5
2
Let’s check this out with simple numbers
6
 2
3
36
9

36
9
4  2
Surds
Higher Mathematics
Rules of surds
We can divide surds
in general
a
b

a
b
Note that this works both ways
a

b
a
b
Surds
Higher Mathematics
A useful tip
a a a
a a
2
 a
2
a
Surds
Higher Mathematics
Examples
Simplify
20
4
24
6
Surds
Higher Mathematics
Rules
a b 
a
b

ab
a
b
Surds
Higher Mathematics
Applications
We can use the rules of surds for simplification
Simplify:
Look for largest square factor
20
45
4 5
2 5
Surds
Higher Mathematics
Applications
We can use the rules of surds for simplification
Simplify:
Look for largest square factor
50
25  2
25 2
5 2
Surds
Higher Mathematics
Applications
We can use the rules of surds for simplification
Simplify:
Look for largest square factor
18  2
92  2
9 2 2
3 2 2
 2 2
Surds
Higher Mathematics
Applications
We can use the rules of surds for simplification
Simplify:
Look for largest square factor
75  12
25  3  4  3
25 3  4 3
5 32 3
3 3
Surds
Higher Mathematics
Applications
We can use the rules of surds for simplification
Simplify:
3  15
3  15
45

95
9 5
 3 5
Surds
Higher Mathematics
Applications
We can use the rules of surds for simplification
Simplify:
Use FOIL
 2  2 3  2 
62 2 3 2  2 2
65 2 2
85 2
Surds
Higher Mathematics
Applications
We can use the rules of surds for simplification
Simplify:
 1
 1


2

2



2
2



Use FOIL
1
1
1


 2 
2
2
2

1

2 2
2 2

1
2
4
2
1

2
2 2
 
3
2
Surds
Higher Mathematics
Applications
Rationalise the denominator and simplify where possible:
2
6
To get rid the surd – multiply top and bottom by the surd
2

6
6
6
2 6

6

6
3
Surds
Higher Mathematics
Applications
Rationalise the denominator and simplify where possible:
20
5
To get rid the surd – multiply top and bottom by the surd
20

5
5
5
20 5

5
 4 5
Surds
Higher Mathematics
Applications
Rationalise the denominator and simplify where possible:
1
2 3
To get rid the surd – multiply top and bottom by the conjugate
conjugate - the same expression with the opposite sign in the middle
1
2 3

2 3 2 3

2 3
 2  3  2  3 

2 3
42 3 2 3  3 3
 2 3
Surds
Higher Mathematics
Applications
Rationalise the denominator and simplify where possible:
4
5 1
To get rid the surd – multiply top and bottom by the conjugate
conjugate - the same expression with the opposite sign in the middle
4
5 1

5 1
5 1

4


5 1
5 1

4


5 1
4
 1 5
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