1. Surds
Simplifying a Surd
Rationalising a Surd

2. Starter Questions 1. = 6 2. = 12 4. = 2 3. = 2 5. 6. =2.76 =1.41
Use a calculator to find the values of : 1.
= 6 2.
= 12 4.
= 2 3.
= 2 5. 6.
=2.76
=1.41

3. We can describe numbers by the following sets:
Surds
We can describe numbers by the following sets:
N = {natural numbers}
= {1, 2, 3, 4, ……….}
W = {whole numbers}
= {0, 1, 2, 3, ………..}
I = {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/ = -7/ = 6/10 = 3/5
55% = 55/100 = 11/20 etc

4. We should also note that
Surds
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

5. 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 π ,√2, 3√5 etc

6. What is a Surd? = 12 = 6 Surds The above roots have exact values
and are called rational
These roots do NOT have exact values
and are called irrational OR
Surds

7. Adding & Subtracting Surds
Note :
√2 + √3 does not equal √5
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.

8. First Rule List the first 10 square numbers
Examples
List the first 10 square numbers
1, 4, 9, 16, 25, 36, 49, 64, 81, 100

9. 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:
To simplify √12 we must split 12 into factors with at least one being a square number.
√12
= √4 x √3
Now simplify the square root.
= 2 √3

10. Have a go √ 45 √ 32 √ 72 = √9 x √5 = √16 x √2 = √4 x √18 = 3√5 = 4√2
Think square numbers
√ 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 10

11. What Goes In The Box ? Simplify the following square roots: (2) √ 27
(3) √ 48
(1) √ 20
= 2√5
= 3√3
= 4√3
(6) √ 3200
(4) √ 75
(5) √ 4500
= 5√3
= 30√5
= 40√2 11

12. First Rule
Examples

13. Have a go
Think square numbers 13

14. Have a go
Think square numbers

15. Exact Values

16. Starter Questions
Simplify : 1. 2.
= 2√5
= 3√2 3. 4.
= ¼
= ¼

17. Second Rule
Examples

18. 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:

19. Rationalising Surds This will help us to rationalise a surd fraction
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
This will help us to rationalise a surd fraction

20. Rationalising Surds
To rationalise the denominator multiply the top and bottom of the fraction by the square root you are trying to remove:
( √5 x √5 = √ 25 = 5 )

21. Rationalising Surds Let’s try this one :
Remember multiply top and bottom by root you are trying to remove

22. Rationalising Surds
Rationalise the denominator

23. What Goes In The Box ?
Rationalise the denominator of the following :

24. Looks something like the difference of two squares
Rationalising Surds
Conjugate Pairs.
Look at the expression :
This is a conjugate pair. The brackets are identical apart from the sign in each bracket .
Multiplying out the brackets we get : =
√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 )

25. Third Rule
Conjugate Pairs.
Examples
= 7 – 3 = 4
= 11 – 5 = 6

26. Rationalising Surds Conjugate Pairs.
Rationalise the denominator in the expressions below by multiplying top and bottom by the appropriate conjugate:

27. Rationalising Surds Conjugate Pairs.
Rationalise the denominator in the expressions below by multiplying top and bottom by the appropriate conjugate:

28. What Goes In The Box?
Rationalise the denominator in the expressions below :
Rationalise the numerator in the expressions below :