1. www.mathsrevision.com Surds Simplifying a Surd Rationalising a Surd www.mathsrevision.com S4 Credit

2. www.mathsrevision.com Starter Questions Use a calculator to find the values of : = 6= 12 = 2 S4 Credit

3. www.mathsrevision.com Learning Intention Success Criteria 1.To explain what a surd is and to investigate the rules for surds. 1.Learn rules for surds. The Laws Of Surds 1.Use rules to simplify surds. S4 Credit

4. www.mathsrevision.com Surds N = {natural numbers} = {1, 2, 3, 4, ……….} W = {whole numbers} = {0, 1, 2, 3, ………..} Z = {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 / 1 -7 = -7 / 1 0.6 = 6 / 10 = 3 / 5 55% = 55 / 100 = 11 / 20 etc We can describe numbers by the following sets: S4 Credit

5. www.mathsrevision.com 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 Surds S4 Credit

6. www.mathsrevision.com Surds QZWN 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 ,  2, 3  5 etc R

7. What is a Surd = 6 = 12 The above roots have exact values and are called rational These roots do NOT have exact values and are called irrational OR Surds S4 Credit

8. www.mathsrevision.com 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. Note : √2 + √3 does not equal √5 S4 Credit

9. www.mathsrevision.com First Rule List the first 10 square numbers Examples 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 S4 Credit

10. www.mathsrevision.com 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:  12 To simplify  12 we must split 12 into factors with at least one being a square number. =  4 x  3 Now simplify the square root. = 2  3 S4 Credit

11. www.mathsrevision.com  45 =  9 x  5 = 3  5  32 =  16 x  2 = 4  2  72 =  4 x  18 = 2 x  9 x  2 = 2 x 3 x  2 = 6  2 Have a go ! Think square numbers S4 Credit

12. www.mathsrevision.com What Goes In The Box ? Simplify the following square roots: (1)  20 (2)  27(3)  48 (4)  75(5)  4500 (6)  3200 = 2  5= 3  3 = 4  3 = 5  3 = 30  5= 40  2 S4 Credit

13. www.mathsrevision.com First Rule Examples S4 Credit

14. www.mathsrevision.com Have a go ! Think square numbers S4 Credit

15. www.mathsrevision.com Have a go ! Think square numbers S4 Credit

16. www.mathsrevision.com 8-Aug-15Created by Mr Lafferty Maths Dept Now try MIA Ex 7.1 Ex 8.1 Ch9 (page 185) S4 Credit Exact Values

17. www.mathsrevision.com Starter Questions Simplify : = 2√5= 3√2 = ¼ S4 Credit

18. www.mathsrevision.com Learning Intention Success Criteria 1.To explain how to rationalise a fractional surd. 1.Know that √a x √a = a. The Laws Of Surds 2.To be able to rationalise the numerator or denominator of a fractional surd. S4 Credit

19. www.mathsrevision.com Second Rule Examples S4 Credit

20. www.mathsrevision.com 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: S4 Credit

21. www.mathsrevision.com Rationalising 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”. Remember the rule This will help us to rationalise a surd fraction S4 Credit

22. www.mathsrevision.com 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 ) Rationalising Surds S4 Credit

23. www.mathsrevision.com Let’s try this one : Remember multiply top and bottom by root you are trying to remove Rationalising Surds S4 Credit

24. www.mathsrevision.com Rationalising Surds Rationalise the denominator S4 Credit

25. www.mathsrevision.com What Goes In The Box ? Rationalise the denominator of the following : S4 Credit

26. www.mathsrevision.com Conjugate Pairs. Rationalising Surds 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 ) S4 Credit Looks something like the difference of two squares

27. www.mathsrevision.com Third Rule Examples Conjugate Pairs. = 7 – 3 = 4 = 11 – 5 = 6 S4 Credit

28. www.mathsrevision.com Rationalise the denominator in the expressions below by multiplying top and bottom by the appropriate conjugate: Conjugate Pairs. Rationalising Surds S4 Credit

29. www.mathsrevision.com Rationalise the denominator in the expressions below by multiplying top and bottom by the appropriate conjugate: Conjugate Pairs. Rationalising Surds S4 Credit

30. www.mathsrevision.com What Goes In The Box Rationalise the denominator in the expressions below : Rationalise the numerator in the expressions below : S4 Credit

31. www.mathsrevision.com 8-Aug-15Created by Mr Lafferty Maths Dept Now try MIA Ex 9.1 Ex 9.1 Ch9 (page 188) S4 Credit Rationalising Surds