Expanding Brackets with Surds and Fractions Slideshow 9, Mr Richard Sasaki, Room 307

Objectives Be able to expand brackets with surds Expanding brackets with surds on the outside Calculate with surds in fractions

Expanding Brackets (Linear) Let’s think back to algebra. When we expand brackets, we multiply terms on the inside by the one on the outside. 3𝑥 2𝑥−𝑦 = 6 𝑥 2 −3𝑥𝑦 The same principles apply with surds. 2( 2 −3)= 2 2 −6 In this case, the expression cannot be simplified. But sometimes we are able to.

Expanding Brackets (Linear) Let’s try an example where we can simplify. Example Expand and simplify 4(2 3 + 12 ). 4 2 3 + 12 = 8 3 +4 12 =8 3 +4∙2∙ 3 =8 3 +8 3 =16 3 Note: We could simplify initially but then there would be no need to expand.

32 2 20 11 ±5+6 5 ±10− 5 8 3 +40 2 ±28+14 3 ±10+ 15 ±7+2 7 + 14 4 6 +2 2 5 6 −18 2 ±14−4 7 ±11−2 11 ±240−45 2 ±6+2 3 ±3− 6 36 70 +18 10 +12 2

Multiplying Surds Remember, when we multiply a surd by itself, we will end up with a plus or minus number. 3 × 3 =±3 But in actual fact, if we square a surd…it will always be positive. 3 2 =3 Can you see how these two things are different? Anyway, it’s safest to always write ‘±’ symbols for some number 𝑥∈ℚ. Note: If you say 3 × 3 =3, this is acceptable.

Surds in Fractions We had a look at some surd fractions in the form 𝑎 𝑏 𝑐 𝑑 where 𝑎, 𝑏, 𝑐, 𝑑∈ℤ (𝑐, 𝑑≠0). Let’s review. Example Simplify 1 2 3 . 3 2 3 ∙ 3 = 3 6 1 2 3 = Remember, a fraction should have an integer as its denominator.

Surds in Fractions Questions with different denominators require a different thought process. We need to expand brackets. Example Simplify 4 3 +2 3 − 2 7 −5 4 . 4 3 +2 3 − 2 7 −5 4 = 4(4 3 +2) 3∙4 − 3(2 7 −5) 4∙3 = 16 3 +8 12 − 6 7 −15 12 = 16 3 +8−6 7 +15 12 = 16 3 −6 7 +23 12

Answers – Easy – Questions 1 - 5 5 +2 3 +7 4 2 7 −6 3 +21 6 9 2 −4 7 −4 5 +27 12 4 6 +9 6 23 5 +3 6

Answers – Easy – Questions 6-10 7 3 −5 7 +73 35 35 3 −8 6 28 95 3 +42 2 6 11 5 −9 6 27 11 44

Answers – Hard – Questions 1 - 5 2 5 +5 3 5 5 14 +2 6 4 17 3 −69 5 3 651 2 −82 10 35 4 15 −4 3 3

Answers – Hard – Questions 6 - 10 13 2 −4 72 3 −6 2 +12 3 4 3 +9 2 −6 18 6 5 +63 2 −160 30 70 5 −63 7 −15 3 −422 105 (this is the positive root)