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38 Unit 2 | 13 Index notation, BIDMAS and further factorising | Student Book chapter 8 13 Index notation, BIDMAS and further factorising BIDMAS F Work out: a 63713 b 30 2 5 3 2 c 12 4 (4 1 2) a 6 3 7 1 3 5 42 1 3 5 45 Multiplication comes before Addition in BIDMAS. b 30 2 5 3 2 5 30 2 10 5 20 Multiplication must be done BEFORE Subtraction. c 12 4 (4 1 2) 5 12 4 6 5 2 Brackets must be done first. Then the Division can be done. Working with powers C Simplify these expressions by writing as a single power: a 53 3 54 4 52 b 3x5y 3 5x3y7 When multiplying, ADD the powers; when dividing, SUBTRACT the powers. a 53 3 54 4 52 5 53 1 4 — 2 5 55 First rearrange as shown: numbers then x terms then y terms. b 3x5y 3 5x3y7 5 3 3 5 3 x513 3 y117 It is a common error NOT to cube the 2 as well, giving 6x3. c (2x)3 5 15x y 8 8 Now use the rules for indices. c (2x)3 5 2x 3 2x 3 2x 5 8x3 Cubing means repeat the bracket three times and multiply. Expanding brackets Expand: a 3x(2x 1 1) b 7x(8x 2 3y) a 3x(2x 1 1) 5 3x 3 2x 1 3x 3 1 5 6x2 1 3x b 7x(8x 2 3y) 5 56x2 2 21xy Do not forget to multiply BOTH terms by 3x. Multiply each term by the 7 AND the x. Expanding and simplifying Expand and simplify 5(4x 1 y) 2 2(3x 2 2y) Remember that a MINUS outside the bracket changes the sign inside it. 5 20x 1 5y 2 6x 1 4y 5 14x 1 9y First do 20x 2 6x and then 5y 1 4y Factorising Factorise fully: a 6x2 2 3x b 7x2 1 14xy a 6x2 2 3x 5 3x(2x 2 1) 3 323 x 3x1 3 3 x The 3 and the x are common. b 7x2 1 14xy 5 7x(x 1 2y) Take out the common 7x and put in the bracket what is left. Always check to see if when you expand the answer you get back to the start. E.g. in your head expand 7x(x 1 2y) 5 7x2 1 14xy Yes! Unit 2 | 13 Index notation, BIDMAS and further factorising | Student Book chapter 8 Now try these 39 Questions on this spread are targeted at the grades indicated. F 1 Work out: a 24 4 4 1 2 b 8 1 2 3 5 c (23 2 3) 4 (4 1 6) d 7 1 3 3 2 1 4 e 9 2 (6 1 2) f 15 4 (3 1 2) g (6 2 1)2 h 6 1 9 4 3 i 24 4 (3 3 2) E 2 Insert brackets into the following to make each statement true: a 3 1 6 4 2 1 7 5 13 b 60 1 30 4 3 1 2 5 66 c 6 1 3 3 7 2 4 5 27 d 56 2 16 4 4 3 2 5 20 D 3 Penny says that 7(2x 2 3) 5 14x 2 3 Explain why Penny is incorrect. C 4 Simplify these expressions by writing as a single power: a 75 3 74 b 912 4 93 c 53 3 57 4 52 d 69 4 6 e 310 4 35 f 824 3 8 g 220 4 24 h 49 4 43 3 42 5 Simplify the following: a a7 3 a3 b b8 4 b2 c c 3 c7 4 c2 d d15 4 d5 e e9 4 e3 f f 9 3 f 3 f 2 g g24 4 g4 h h14 4 h2 3 h3 3 k5 i ______ k 32 k m20 j _______ 4 m 3m n7 3 n8 k _______ n3 3 n2 p 3 p9 l ______ p3 6 Simplify the following: a 4a8 3 3a7 b 12b7 4 3b2 c 5c5 3 6c6 4 3c3 d 9d18 4 3d6 e (3e8)3 g (7g6)2 30h18 h ________ 2h 3 3h2 2f 5 3 6f 10 f _________ 4f 3 7 Expand: a a(a 1 5) b b(7 2 2b) c c(3c 1 1) d 2d(7 2 d) e 5e(4e 1 9) f 6f(4 2 7f) 8 Expand and simplify: a 4(a 1 5) 1 2(3a 1 7) b 9(2b 1 5) 2 3(6b 1 3) c 8(6 1 c) 2 6(1 2 2c) d 7(8d 2 1) 1 2(9 2 2d) e 5(9e 2 2) 2 10(6e 2 1) 9 Factorise fully: a 8a 1 2 b 9 2 3b d 10 1 15d e ef 1 3e g k 1 7k h 3m 2 m i 8n2 1 2n j 9p 2 15p2 k 7q2r 1 14qr l t2u2 2 5tu 2 c 6c 2 12 f 8g 2 3gh 2