1. Revision - Surds Production by I Porter 2009 0 1 2 1 1 1 1

2. 2 Definition : If p and q are two integers with no common factors and, than a is an irrational number. Irrational number of the form are called SURDS. 0 1 2 1 1 1 1

3. 3 Prove that is irrational. [Reproduction of Proof not required for assessment] Let p and q be two integers with no common factors, such that is a rational number. Then squaring both sides, rearrangeNow, the left side is an even number, this implies that p 2 is also an even number, which implies that p is divisible by 2. Let p = 2m, where m is an integer. Hence, p 2 = 4m 2. 2q 2 = 4m 2 Dividing by 2. q 2 = 2m 2 Now, the right side is an even number, this implies that q 2 is also an even number, which implies that q is divisible by 2. But, this contradict our assumption that p and q have NO common factors.. Hence, we cannot write, were p and q are integers with no common factors. Therefore cannot be a rational number, it must be an IRRATIONAL number..

4. 4 Surd Operations If a, b, c and d are numbers and a > 0 and b > 0, then Surds can behave like numbers and/or algebra.

5. 5 Simplifying Surds To simplify a surd expression, we need to (if possible) write the given surd number as a product a perfect square, n 2, and another factor. It is important to use the highest perfect square factor but not essential. If the highest perfect square factor is not used first off, then the process needs to be repeated (sometimes it faster to use a smaller factor). Generate the perfect square number: Order 12345678910 n2n2 12122 3232 4242 5252 6262 7272 8282 9292 10 2 Value 149162536496481100 Factorise the surd number using the largest perfect square : 18 = 9 x 324 = 4 x 627 = 9 x 348 = 16 x 3 Or any square factor: 72 = 36 x 272 = 9 x 872 = 4 x 1872 = 4 x 9 x 2 Perfect Squares

6. 6 Examples: Simplify the following surds. N 2 1 4 9 16 25 36 49 64 81 100 121 144 169 196 225

7. 7 Exercise: Simplify each of the following surds. N 2 1 4 9 16 25 36 49 64 81 100 121 144 169 196 225

8. 8 Simplifying Surd Expressions. Examples: Simplify the following N 2 1 4 9 16 25 36 49 64 81 100 121 144 169 196 225

9. 9 Exercise: Simplify the following. N 2 1 4 9 16 25 36 49 64 81 100 121 144 169 196 225

10. 10 Simplifying Surd Expressions. Examples: Simplify the following

11. 11 Exercise: Simplify each of the following (fraction must have a rational denominator).

12. 12 Using the Distributive Law - Expanding Brackets. Examples: Expand and simplify the following

13. 13 Exercise: Expand the following (and simplify) Special cases - (Product of conjugates) Difference of two squares.

14. 14 Rationalising the Denominator - Using Conjugate Examples: In each of the following, express with a rational denominator. [more examples next slide]

15. 15 More examples

16. 16 Exercise: Express the following fraction with a rational denominator. Harder Type questions. Express the following as a single fraction with a rational denominator. Example A possible method to solve this problems is to RATIONALISE the denominators of each fraction first, then combine the results.

17. 17 Exercise: Express the following as a single fraction with a rational denominator.