2 7.1 Collecting like terms 1 Simplify a a + a + a + a... b 3b + 2b... c c + 2c + 3c... d 4d + 2d + d + 3d... e 5e 3e... f 3f f... g 7g + 3g 8g... h 2hk + 8hk 3hk... i 3m 2 + m 2... j 9n 2 2n 2 n 2... 2 Simplify a 4p + 2p + 5q + q... b 3r + 4r + 7s 2s... c t + u + t + 3u... d 9v + 3w 5v 2w... e x + 4y + 3x 2y... f 3a 9b a + 5b... g 2c 3d + 8c 5d... h e + 3f 3e + 2f + 1... i 2g h + 4 3g + 7h 5... j 9j + k 2m 5j + 7 3m... 3 The diagram shows a line made from three sections. The length of each section is given in centimetres. 5a 3b a + 7b 4b 2a Write down an expression, in terms of a and b, for the total length of this line....... 107A
2 7.1 Collecting like terms 4 Melissa, Mason and Zach are judges in a talent competition. D a For the first performer: Melissa gave a score of x points. Mason gave three times as many points as Melissa. Zach gave 10 points fewer than Mason. Write down an expression in terms of x for the total points scored by the first performer. b For the second performer: Melissa gave a score of 2x points. Mason gave a score of 50 points. Zach gave x points more than Mason. Write down an expression in terms of x for the total points scored by the second performer. c Write down an expression in terms of x for the difference in points between the first performer and the second performer. 107B
2 7.2 Using substitution 1 Work out the value of each of these expressions when x = 3. a x + 4... b 5x... c x 10... d 2x 9... e 7 + x... f 11 5x... g x 2... h 2x 2... 2 Work out the value of each of these expressions when p = 5 and q = 2. a p + q... b p q... c q p... d 2p + 3q... e 5p 4p... f 6 q p... g 3q p... h p 2 + q 2... i 3q 2... j 10 q 3... 3 Work out the value of each of these expressions when x = 4 and y = 3 and z = 1. D a xy... b yz... c 2xyz... d 2xy + 3yz... e 5xy yz... f x 2 + y 2 + z 2... g 4xy 2... h z 3... i 5z 10... j 6x 2 yz 3... 109
2 7.3 Using the index laws 1 Simplify a a a a a... b 3c 6c c... D c 2e 2e... d f 2f 3f... 2 Simplify Add indices when multiplying. C a g 3 g 2... b k 4 k... c m m 3 m 5... d p p p... 3 Simplify Multiply numbers then letters. a 3q 3 2q 5... b s 4 4s 3... c 4u 2 3u 3 2u... d v 8 2v 4 3v 2... 4 Simplify a 2xy 2 3x 2 y... b 4x 5 y 2 x 3 y 2... c mn 3 2m 3 n 5m 2 n 2... d 10x 2 y 3 z 10xy 4 z 3... 5 Simplify Subtract indices when dividing. a a 5 a 2... b b 6 b... 9 c c 3... d e c e 7 2... 111A
2 7.3 Using the index laws 6 Simplify Divide numbers then letters. C a 12f 5 6f 2... b 30g 8 12g... c 35 10 h 4 d 8 2 k 5 7h... 2k... 7 Simplify a 24a 3 b 8 8ab 5... b 100pq 3 20pq... c 15 2 9 xy 5 3 d 4 4 9 3 c c 5 5 xy... 12c... 8 Simplify Multiply indices when raising one power to another. a (x 3 ) 2... b (y 4 ) 7... c (w 5 ) 4... d (z 2 ) 10... 9 Simplify Remember to deal with numbers as well as letters. a (3p 2 ) 4... b (2q 5 ) 6... c (10x 6 ) 3... d 4... y 3 2 111B
2 7.3 Using the index laws 10 Simplify B a (3xy 4 ) 3... b (5e 2 f 4 ) 4... 3 5 3 2 zb d 4 3 zb... c (10p 2 q 6 ) 3... 111C
2 7.4 Fractional and negative powers 1 Simplify B a x 1... b (y 3 ) 1... c z 2... d w 3... 2 Simplify a (a 4 ) 2... b (b 3 ) 5... c (x 3 ) 2... d (y 4 ) 6... 3 Simplify A a (x 5 ) 0... b (3x 7 y) 0... c (2pq 5 ) 1... d (10c 2 d 5 ) 2... 4 0 2 x e 3 10 x f... 3 2 2a b 5 8 4ab 1... 113A
2 7.4 Fractional and negative powers 4 Simplify A a (x 8 ) 1 2... b (y 10 ) 1 4... c (16x 6 ) 1 2... d (25y 6 ) 1 4... e (125w 6 x 12 ) 1 3... f (9x 8 y 7 ) 1 2... 5 Simplify a (a 2 ) 1 3... b (c 6 ) 1 3... c (32x 15 ) 1 5... d (x 2 y 6 ) 1 2... 113B
2 7.5 Term-to-term and position-to-term definitions Find a the rule, b the next two terms, and c the 12th term for each of the following number sequences. 1 1 3 5 7 2 2 4 6 8 3 0 4 8 12 4 4 7 10 13 5 3 2 7 12 6 28 25 22 19 7 120 113 106 99 8 2 8 18 32 9 0 2 6 12 10 1 6 17 32 11 If the rule for a sequence is add 4 and the 5th term is 19, find the a 1st term b 10th term. 115
2 7.6 The nth term of an arithmetic sequence Write down a the difference between consecutive terms, and b the zero term for each of the following arithmetic sequences. 1 1 3 5 7 C a... b... 2 12, 7 2 3 a... b... 3 21 18 15 12 a... b... 4 Here are the first five terms of an arithmetic sequence 3, 7, 11, 15, 19 a Write down, in terms of n, an expression for the nth term of this arithmetic sequence. b Use your answer to part a to work out the i 15th term,... ii 75th term.... 5 Here are the first four terms of an arithmetic sequence 6, 10, 14, 18 a Write down, in terms of n, an expression for the nth term of this arithmetic sequence. b Use your answer to part a to work out the i 12th term,... ii 20th term.... 6 Here are the first four terms of an arithmetic sequence 124, 118, 112, 106 Explain why the number 7 cannot be a term of this sequence. 117A
2 7.6 The nth term of an arithmetic sequence 7 Here are the first five terms of an arithmetic sequence. 2 9 16 23 30 Write down, in terms of n, an expression for the nth term of this sequence. C a Joanna says that 344 is a term in this sequence. Is Joanna right or wrong? You must fully explain your answer. b. 8 Here are the first five terms of two arithmetic sequences. 3 11 19 27 35 0 3 6 9 12 Show that the number 315 is a term in both sequences.... 117B