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Surds

S4 Credit. Surds. Simplifying a Surd. Rationalising a Surd. www.mathsrevision.com. S4 Credit. Starter Questions. Use a calculator to find the values of :. = 6. = 12. = 3. = 2. The Laws Of Surds. S4 Credit. Learning Intention. Success Criteria.

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Surds

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

  2. S4 Credit Starter Questions Use a calculator to find the values of : = 6 = 12 = 3 = 2 www.mathsrevision.com

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

  4. Surds S4 Credit We can describe numbers by the following sets: = {1, 2, 3, 4, ……….} N = {natural numbers} 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

  5. Surds S4 Credit 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

  6. Surds N W Z Q R 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

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

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

  9. First Rule S4 Credit Examples List the first 10 square numbers 1, 2, 4, 9, 16, 25, 36, 49, 64, 81, 100 www.mathsrevision.com

  10. Simplifying Square Roots S4 Credit Some square roots can be broken down into a mixture of integer values and surds. The following examples will illustrate this idea: To simplify 12 we must split 12 into factors with at least one being a square number. 12 = 4 x 3 Now simplify the square root. = 2 3 www.mathsrevision.com

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

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

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

  14. Have a go ! Think square numbers S4 Credit

  15. Have a go ! Think square numbers S4 Credit

  16. Exact Values S4 Credit Now try MIA Ex 7.1 Ex 8.1 Ch9 (page 185) Created by Mr Lafferty Maths Dept

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

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

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

  20. Rationalising Surds S4 Credit 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: www.mathsrevision.com

  21. Rationalising Surds S4 Credit 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 www.mathsrevision.com

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

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

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

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

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

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

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

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

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

  31. Rationalising Surds S4 Credit Now try MIA Ex 9.1 Ex 9.1 Ch9 (page 188) Created by Mr Lafferty Maths Dept

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