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

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.

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

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

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.

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

Solving equations Solve

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

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