Slideshow 10, Mr Richard Sasaki, Mathematics Conjugations in Surds Slideshow 10, Mr Richard Sasaki, Mathematics

Objectives Be able to rationalise denominators of fractions in the form 𝑎 +𝑏 Do the same for denominators in the form 𝑎 𝑏 +𝑐 Simplify expressions with surds in this form

Rationalising the Denominator Previously, we learned how to make the denominator an integer on a fraction. 1∙ 2 2 ∙ 2 = 2 2 1 2 = If the denominator is in the form 𝑎 𝑏 , we can multiply the top and bottom by 𝑏 as 𝑎 𝑏 ∙ 𝑏 =𝑎𝑏. How about denominators in the form 𝑎 +𝑏? We need to make a conjugation. What’s that?

Conjugation In maths (not English), a conjugation refers to things that link to one another (a relationship). The conjugate of 𝑎 +𝑏 is . 𝑎 −𝑏 Example Simplify 1 2 +1 . 1 2 +1 = 1 2 +1 ∙ 2 −1 2 −1 = 2 −1 2 2 − 1 2 = 2 −1 2−1 = 2 −1

Conjugation Example Simplify 3 2 5 −2 . 3 2 5 −2 = 3 2 5 −2 = 3 2 5 −2 ∙ 5 +2 5 +2 = 3 2 ∙( 5 +2) 5 2 − 2 2 = 3 10 +6 2 5−4 = 3 10 +6 2

2 −3 5 +2 3 +1 3 −1 2 5 +2 −2 2 −3 −2 3 +8 13 10 + 2 4 3 +3 4 −5 3 −3 11 −54 7 −42 37 6 6 −12 −7 10 −35 2 20 135 2 −18 223 −500 6 −600 19

Answers – Part 1, Hard 12−6 2 4 2 −2 6 5 +2 30−5 2 17 3− 3 2 18 2 −18 12−6 2 4 2 −2 6 5 +2 30−5 2 17 3− 3 2 18 2 −18 4 10 21 −6 5 4 15 −3 5 3 6 + 3 − 2 +2

Other Types When we need to conjugate a denominator in the form 𝑎 𝑏 +𝑐, we multiply the numerator and denominator by . 𝑎 𝑏 −𝑐 Example Simplify 3 5 2 3 −5 . 3 5 2 3 −5 = 3 5 2 3 −5 ∙ 2 3 +5 2 3 +5 = = 3 5 ∙(2 3 +5) 2 3 2 − 5 2 = 6 15 +15 5 12−25 = −6 15 −15 5 13

Answers – Part 2, Easy 3 2 +1 9 3 −5 5 7 −2 3 2 +4 2 5 3 −2 71 3 2 +1 9 3 −5 5 7 −2 3 2 +4 2 5 3 −2 71 7 5 +1 732 −20 2 +36 31 − 6 −3 3 21 4 21 +3 7 39 5 3 +75 74 56 15 +42 5 39 5 5 +50 76 18−4 2 73 21 2 +6 47 −2 7 −1 27

3 + 2 7 − 5 2 3 −3 2 5 + 3 2 7 +2 3 5 4 2 − 6 26 3 42 −14 13 3 − 2 6 +3 48 2 +220 119 −781 35 −1463 1972