1. The Laws Of Surds.

2. 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

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. 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”.

5. 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

6. What Goes In The Box ? 2.
Rationalise the denominator of the following expressions:

7. 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 =
= -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.

8. 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.

9. What Goes In The Box ? 3.
Rationalise the denominator in the expressions below :
Rationalise the numerator in the expressions below :