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Surds Simplifying a Surd Rationalising a Surd Conjugate Pairs Starter Questions Use a calculator to find the values of : 1. 3. 5. 3 36 = 6 2. 8 144 = 12 4. 4 16 2 1.41 6. 3 21 2.76 =3 =2 The Laws Of Surds Learning Intention 1. To explain what a surd is and to investigate the rules for surds. Success Criteria 1. Learn rules for surds. 1. Use rules to simplify surds. What is a Surd 144 = 12 36 = 6 The above roots have exact values and are called rational 2 1.41 3 21 2.76 These roots do NOT have exact values and are called irrational OR Surds Note : Adding & Subtracting √2 +Surds √3 does not equal √5 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. 4 2+6 2 16 23 - 7 23 =10 2 10 3 + 7 3 - 4 3 =9 23 =13 3 First Rule a b ab Examples 4 6 24 4 10 40 List the first 10 square numbers 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 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 = 4 x 3 = 2 3 To simplify 12 we must split 12 into factors with at least one being a square number. Now simplify the square root. Have a go ! Think square numbers 45 32 72 = 9 x 5 = 16 x 2 = 4 x 18 = 35 = 42 = 2 x 9 x 2 = 2 x 3 x 2 = 62 What Goes In The Box ? Simplify the following square roots: (1) 20 (2) 27 (3) 48 = 25 = 33 = 43 (4) 75 (5) 4500 (6) 3200 = 53 = 305 = 402 Starter Questions Simplify : 1. 3. 20 = 2√5 2. 1 1 = 2 2 ¼ 18 = 3√2 1 1 4. = 4 4 ¼ The Laws Of Surds Learning Intention 1. To explain how to rationalise a fractional surd. Success Criteria 1. Know that √a x √a = a. 2. To be able to rationalise the numerator or denominator of a fractional surd. Second Rule a a a Examples 4 4 4 13 13 13 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. 2 numerator = 3 denominator Fractions can contain surds: 2 3 5 4 7 3 2 3- 5 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 a a a This will help us to rationalise a surd fraction Rationalising Surds To rationalise the denominator multiply the top and bottom of the fraction by the square root you are trying to remove: 3 3 5 = 5 5 5 3 5 = 5 ( 5 x 5 = 25 = 5 ) Rationalising Surds Let’s try this one : Remember multiply top and bottom by root you are trying to remove 3 3 7 3 7 3 7 = = = 14 2 7 2 7 7 2 7 Rationalising Surds Rationalise the denominator 10 10 5 10 5 2 5 = = = 7 5 7 5 5 7 5 7 What Goes In The Box ? Rationalise the denominator of the following : 7 3 4 9 2 7 3 = 3 2 2 9 4 6 2 6 = 3 14 3 10 = 2 5 7 3 2 15 = 21 6 3 11 2 3 6 = 11 7 10 15 Starter Questions Conjugate Pairs. Multiply out : 1. 3 3= 3 2. 14 14 = 14 3. 12 + 3 12 - 3 = 12- 9 = 3 The Laws Of Surds Conjugate Pairs. Learning Intention 1. To explain how to use the conjugate pair to rationalise a complex fractional surd. Success Criteria 1. Know that (√a + √b)(√a - √b) = a - b 2. To be able to use the conjugate pair to rationalise complex fractional surd. Looks something like the difference of two squares Rationalising Surds Conjugate Pairs. Look at the expression : ( 5 2)( 5 2) This is a conjugate pair. The brackets are identical apart from the sign in each bracket . Multiplying out the brackets we get : ( 5 2)( 5 2) = 5 x5 - 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 ) Rationalising Surds Conjugate Pairs. Rationalise the denominator in the expressions below by multiplying top and bottom by the appropriate conjugate: 2 5 -1 2( 5 + 1) = ( 5 - 1)( 5 + 1) 2( 5 + 1) 2( 5 + 1) = = ( 5 5 - 5 + 5 - 1) (5 - 1) ( 5 + 1) = 2 Rationalising Surds Conjugate Pairs. Rationalise the denominator in the expressions below by multiplying top and bottom by the appropriate conjugate: 7 ( 3 - 2) 7( 3 + 2) = ( 3 - 2)( 3 + 2) 7( 3 + 2) = (3 - 2) = 7( 3 + 2) What Goes In The Box Rationalise the denominator in the expressions below : 5 ( 7-2) 5( 7 + 2) = 3 3 =3+ ( 3 - 2) 6 Rationalise the numerator in the expressions below : 6+4 12 -5 = 6( 6 - 4) 5 + 11 7 -6 = 7( 5 - 11)