Surd Bracket Expansion Slideshow 13 　Mathematics Mr Richard Sasaki

Objectives Review bracket expansion types Be able to square surd expressions in the form 𝑎 𝑏 +𝑐 and 𝑎 𝑏 +𝑐 𝑑 Multiply surd expressions in the form 𝑎 𝑏 +𝑐 and 𝑎 𝑏 +𝑐 𝑑

Review We know how to multiply a pair of binomials. Let’s review the rules. 𝑥+𝑎 2 = 𝑥 2 +2𝑎𝑥+ 𝑎 2 𝑥+𝑎 𝑥−𝑎 = 𝑥 2 − 𝑎 2 𝑥−𝑎 2 = 𝑥 2 −2𝑎𝑥+ 𝑎 2 Let’s use these expressions to multiply surd expressions.

Surd Expressions In chapter one, we multiplied surd expressions in the form 𝑎 𝑏 𝑐 𝑑 +𝑓 𝑔 . 𝑎 𝑏 𝑐 𝑑 +𝑓 𝑔 = 𝑎𝑐 𝑏𝑑 +𝑎𝑓 𝑔𝑏 Example Expand and simplify 2 5 (3 2 +5 6 ). 2 5 3 2 +5 6 = 6 10 +10 30 We know how to do these!

Surd Expressions We also know how to conjugate surds. We used the rule 𝑥+𝑎 𝑥−𝑎 = 𝑥 2 − 𝑎 2 to produce two integers. (𝑎 𝑏 +𝑐 𝑑 )(𝑎 𝑏 −𝑐 𝑑 )= 𝑎 2 𝑏− 𝑐 2 𝑑 Example Write (3 2 +6 11 )(3 2 −6 11 ) as an integer. 3 2 2 − 6 11 2 = 3 2 ∙2− 6 2 ∙11 =18−396 =−378 Note: Remember… ℤ is the set. ℚ is the set. integer rational number

37∈ ℤ + 53∈ ℤ + 291∈ ℤ + 98∈ ℤ + 148∈ ℤ + −1396∈ ℤ − 3 2 2+2 3 =6 6 +6 2 . 6 6 +6 2 can’t be simplified further. ∴3 2 2+2 3 ∉ℤ. 3 3 4 12 −2 27 =12 36 −6 81 = 18∈ℤ. ∴3 3 4 12 −2 27 ∈ℤ. 4−2 5 4+2 5 3 = 4 2 − 2 5 2 3 =− 4 3 ∈ℚ as −4, 3 ∈ℤ. ∴ 4−2 5 4+2 5 3 ∈ℚ.

Squaring Rules for surd expressions have the same rules as multiplying binomials. 𝑥+𝑎 2 = 𝑥 2 +2𝑎𝑥+ 𝑎 2 𝑎 𝑏 +𝑐 2 = 𝑎 2 𝑏+2𝑎𝑐 𝑏 + 𝑐 2 𝑎 𝑏 +𝑐 𝑑 2 = 𝑎 2 𝑏+2𝑎𝑐 𝑏𝑑 + 𝑐 2 𝑑 Example Expand and simplify 3 2 +4 2 . 3 2 +4 2 = 3 2 2 +2∙3 2 ∙4+ 4 2 =9∙2+24 2 +16 =24 2 +34 Note: All roots are positive as everything is squared.

2 2 +3 9−4 5 16 5 +36 259−30 10 29−12 5 985−80 15 4 21 +19 48−6 15 46−12 14 98 12 286 +203 421−28 58

Multiplication Let’s review another rule… (𝑥+𝑎)(𝑦+𝑏)= 𝑥𝑦+𝑎𝑦+𝑥𝑏+𝑎𝑏 This is literally just gathering the combinations. (𝑎 𝑏 +𝑐 𝑑 )(𝑚 𝑛 +𝑝 𝑞 )= 𝑎𝑚 𝑏𝑛 +𝑐𝑚 𝑑𝑛 +𝑎𝑝 𝑏𝑞 +𝑐𝑝 𝑑𝑞 Example Expand and simplify 2 5 −3 2 7 +5 . 2 5 −3 2 7 +5 =2 5 ∙2 7 −3∙2 7 +2 5 ∙5−3∙5 =4 35 −6 7 +10 5 −15

2 35 +3 7 +2 5 +3 3 14 −2 7 +12 2 −8 48−16 3 21 3 +37 6 7 +54 40−22 5 −2 35 +6 7 +84 5 −1 4 95 +10 19 −4 5 −10 22−6 5 117−113 3

2 14 +2 7 +4 2 +7 4 14 −12 7 +16 2 −21 6 77 +15 11 +4 14 +10 2 26 6 +130 10 6 −15 3 +24 2 −32 −20 6 +7 3 −28 2 +15 −8 21 −36 7 −315 3 −210 −40 182 +40 130 −30 14 +30 10 3616−256 78 −57 6 −11 3 +111