1. Surds Learning objectives Different kind of numbers
Surds and their properties
Difference of two squares
Rationalising the denominators

2. Different kinds of number
Natural numbers : These are the set of numbers: 1, 2, 3, 4, 5, ...,n as used in counting.
It is a matter of choice whether 0 is included or not.
Integers: These are numbers made from the natural numbers by putting a positive or negative sign in front.
Example ... –3, -2, -1, 0, 1, 2, 3,
Rational numbers: A rational number can be expressed as a fraction of two integers:
b cannot be zero
Irrational number has no exact answer and they are expressed as surds: e.g. 2, 3, 5,  etc.

3. Surds and their properties
e.g. 18 = 9 x 2 = 32
e.g. 27 + 75 = 33 + 53 = 83

4. Difference of two squares
a2 – b2 = (a – b)(a + b)
a – b = (a – b)(a + b) 
Examples: Simplify
1. (2 – 1)(2 + 1) =
2 – 1 = 1
2. (5 – 1)(5 + 1) =
5 – 1 = 4
3. (32 – 2)(32 + 2) =
18 – 4= 14
4. (52 – 27)(52 + 27) =
50 – 28 = 22

5. Rationalising the denominators
 If a fraction in the form
then multiply top and bottom of the fraction by b.
 If a fraction is in the from
then multiply the top and the bottom of the fraction
by a  b.
 If a fraction is in the from
then multiply the top and the bottom of the fraction
by a + b.

6. Rationalising the denominators
Examples: rationalize the denominators 1. 2. 3. 4. 5.

7. Solving equations
Solve

8.  Write the expression as the product of two brackets and expand
 Rationalise the denominator and simplify

9.  Substitute into the formula for the area of a triangleC
 Substitute into the formula for the area of a triangle
 Rationalise the denominator to get the required format