1. Surds
Objectives:
Grade A: Rationalise the denominator of a surd, such as
Grade A*: Simplify surds such as write in the form

2. Surds
Prior knowledge: 52
152
132
112 92
√144
√196
√49
Identify factors that are square numbers: 20 32
500 45 72 18 98 80 25
4 × 5
225
16 × 2
169
100 × 5
121
9 × 5 81
36 × 2 12
9 × 2 14
49 × 2 7
16 × 5

3. Surds
Rational number
a rational number is a number which can be expressed
as a ratio of two integers. Such that Where a and b are integers
e.g =
Irrational number
an irrational number is a number that is not an integer and
cannot be expressed as a fraction.
e.g , , , π
Surd
a surd is a number containing an irrational root,
e.g or 3 + 2

4. Surds
Why use surds?
To write an irrational number in surd form is to write it
accurately.
e.g. π is an irrational number, it has an infinite number of decimal places
if you were to write it on paper as digits, at some point you would have
to round it and then it would stop being completely accurate.
We can find out whether a number is rational or irrational by
simplifying it as far as possible.
If we have simplified as far as possible and we are left with a
number such as a square root of a number that is not a natural
square number or a value such as π then we can determine that
the number is irrational

5. Surds
Example:
Are these numbers rational or irrational? √ 5
irrational – because it cannot be simplified to an integer √ 36
rational – because it can be simplified to an integer √
36 = 6 π 2
irrational – because π is irrational π 2π
rational – because it can be simplified to 1 2

6. Surds
When simplifying surds it is useful to know the following √
4 = 2
2 × 2 = 4
therefore √
4 ×
4 = 4
and √
2 ×
2 = 2 √ 16
Also because 4 = √
4 ×
4 = 16
Rule 1 √
a ×
b = ab
Example:
Simplify √ 98
Firstly we will look for factors of 98
that are square numbers
49 × 2 = 98
so can be written as √ 98 √
49 × 2
Using the rule √
49 × 2 √ 2 =7

7. Surds √ 32
Simplify these: 1. 2. 3. 4. 5. 6. 20
500 45 72 18 √
4 × 5 √
4 × 5 = √ 5
= 2 √
16 × 2 √
16 × 2 = √ 2
= 4 √
100 × 5 √
100 × 5 = √ 5
= 10 √
9 × 5 √
9 × 5 = √ 5
= 3 √
36 × 2 √
36 × 2 = √ 2
= 6 √
9 × 2 √
9 × 2 = √ 2
= 3

8. Surds
Rule 2 √ a b √ a b =
Example: √ 96 3
Simplify √ 96 3 =
using rule 2
Find factor(s) of 96 that are square numbers √
16×6 3 = √ 6 3 =
16 × √ 6 3 =
4 ×
6 = 2 × 3 so √
6 =
2 × 3
Rule √
a ×
b = ab so √ 6 3
4 × = √ 3 2×
4 × √ 2 =4

9. Surds
Simplify these: 1. 2. 3. 4. √ 40 5 45 20 81 16 72 √ 8 = √
4 × 2 √
4 × 2 = √ 2
= 2 √ 81 16 = = 9 4
= 2 1 4 √ 9 4 √ 9 4 = = 3 2
= 1 1 2 √ 9 2 √ 9 2 = = 3 √ 2

10. Surds
Simplify these: 5. 6. 7. 8. √ 30 5 8 4 10 6 2 21 3 = √ 10 5
3 × = √ 5 2×
3 × √ 2
= 3 = √ 30 5
2 × = √ 5 6×
2 × = √ 3 2×
2 × √ 2
= 2 3 = √ 3 7× 4 = √ 7 4 = √ 10 2 5 = √ 5 2× 2 = √ 5

11. Surds
Rule 3 √
c +
c = c a b
(a + b)
Example:
Simplify √
2 + 32 3 √
16 × 2
2 + 3 = √
16 × 2
2 + 3 = √ 2 4
2 + 3 = √ 2
= 7

12. Surds √ 2
2 + 3
Simplify these: 1. 2. 3. 4. 5
5 - 2
20 + 28
7 - √ 2
= 4 √ 5 = √
4 × 5 = 5 + √
4 × 5 = + √ 5
= 2 + √ 5
= 3 √
4 × 7 - 7
= 3 √ 7
= 3
4 × - √ 7
= 3
- 2 √ 7 =