B 144 c 1 d e 3 8 f 3 125 g 3 1000 h 3 1 i 3 j 43

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b ; ( ) 144 c ; ( ) 1 d ; ( ) 0 e ; 3 ( ) 8 f ; 3 ( ) 125 g ; 3 ( ) 1000 h ; 3 ( ) 1 i ; 3 ( ) 0 j
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43 Extra Exercises for Practice Evaluate 3 3 75 ( ) 121; ( ) 225; ( ) ; ( ) 216; ( ) 0 3 - a b c d e 2.2 Indices Rules There are generally three rules for indices: 1. Multiplication rule 2. Division rule 3. Power rule 2.2.1 Multiplication Rule Let us start with some examples: 2 2 4 6 2 4 2 4 6 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 + × = × × × × × = × = = E5F E555555F 3 7 3 7 10 3 7 3 7 10 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 + × = × × × × × × × × × = × = = E555F E5555555555555F From the two examples we have just done, we can see that to multiply two powers of the same number we just add up the powers. 2.2.2 Division Rule Again let us try some examples: In general : m n m n a a a + × =
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3 3 3 2 3 2 1 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 1 3 or - - × × = = = = × = = = × 5 5 2 5 3 5 3 2 3 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 1 7 or - - × × × × × = = = = × = = × × 2 2 1 3 3 - = 2.2.3 Power rule As before, we should start by doing some examples: 2 3 3 6 2 3 2 3 6 (5 ) (5 5) (5 5) (5 5) (5 5) 5 5 5 5 5 5 5 (5 ) 5 5 or × = × = × × × × × = × × × × × = = = 3 4 4 12 1 2 3 4 3 4 3 4 12 (2 ) (2 2 2) (2 2 2) (2 2 2) (2 2 2) (2 2 2) 2 2 2 2 2 2 2 2 2 2 2 2 2 (2 ) 2 2 or × = × × = × × × × × × × × × × × = × × × × × × × × × × × = = = E5555F E5555F E5555F E5555F In general : p p q p q q a a a a a - - = × = and 1 - = q q a a In general : ( ) r s r s a a × = Note : In addition to these three laws we also need to know that 0 1 a = , i.e., apart from zero, any number raised to the power zero is equivalent to one (we can check this using our calculators). Exercise 2.2 Evaluate the following : × 2 ( ) 3 3 a ; - × 4 1 ( ) 5 5 b ; × 3 ( ) 10 10 c 3 2 2 ( ) 2 d ; 4 5 4 ( ) 4 e ; 3 10 ( ) 10 f 2 4 ( ) (3 ) g ; 3 2 ( ) (5 ) h ; - 1 2 ( ) (6 ) i
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45 Extra Exercises for Practice Evaluate 3 2 5 2 4 3 3 3 2 6 2 64 ( ) 3 3 ; ( ) 2 8 ; ( ) 10 100 ; ( ) ; ( ) 16 4 - × × × a b c d e
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46 ANSWERS Topic 1 Answer 1.1.1: 2 2 3 2 ( ) 7 2; ( ) 5 3 ; ( ) 3 5 a x b x xy y c x x x + - + - - - - . Answer 1.1.2 : 4 1 9 16 2 ) ; ( ) ; ( ) ; ( ) ; (e) 3 3 2 9 3 a b c d cannot besimplified = . Answer 1.1.3: 2 3 3 3 ( ) ; ( ) 3 ; ( ) ; ( ) ; ( ) a a b x c x d x e x b . Answer 1.2.1: (a) 5 15 x + ; (b) 2 6 x x - ; (c) 15 6 ax a - ; (d) 2 10 21 y y - - ; (e) 2 45 36 x x - + ; (f) ac ad bc bd + + + ; (g) 2 7 10 x x - + . Answer 1.3.1: ( 29 ( 29 ( 29 ( 29 ( 29 ( ) 5 5 ;( ) 9 3 2 ;( ) 8 3 ;( ) 4 ;( ) ; - + - - + + a x b x c x d x x e xy x y ( 29 ( 29 ( ) 4 1 ;( ) 8 3 1 - + f a ax g a ax . Answer 2.1.1 22 9 ( ) 3 ; ( ) 12; ( ) 0; ( ) ; ( ) 36; ( ) 4; ( ) ; ( ) 8;( ) 0; 3 2 - - = a b c d e f g h j 9 ( ) ; ( ) 0 10 - + = k l is not a linear equation because not of the form ax b . Answer 2.2.1: ( ) 8 2 a x or x = - ; ( ) 4 5 b x or x = - = - ; ( ) 3 5 c x or x = = ; ( ) 5 8 d x or x = = - ; ( ) 0 12 e x or x = = . Answer 2.2.2: 1 4 5 1 ( ) ; ( ) 2 1; ( ) 5 7; ( ) 5 5 3 3 a and b and c and d and - - - - - - . Answer 2.2.3: 1 ( ) 4 2 a x and y = = ; ( ) 2 1 b x and y = = ; 1 ( ) 3 2 c x and y = = - ; ( ) 3 3 d x and y = - = . Answer 3.1: same as in 2.2.3 Topic 2 Answer 1.1: 1 11 34 37 37 ( ) ; ( ) ; ( ) ; ( ) ; ( ) 9 12 15 36 36 a b c d e .
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47 Answer 1.2: 5 1 2 1 ( ) ; ( ) ; ( ) ; ( ) ; ( ) 7; ( ) 0 8 6 35 4 a b c d e f - = . Answer 1.3: 8 10 3 15 9 9 ( ) ; ( ) ; ( ) ; ( ) ; ( ) ; 3 9 2 13 2 0 - - - ÷ a b c d e impossible ( ); f impossible . Answer 2.1: 8 10 3 15 9 9 ( ) ; ( ) ; ( ) ; ( ) ; ( ) ; 3 9 2 13 2 0 - - - ÷ a b c d e impossible ( ); f impossible . Answer 2.2: 7 3 4 1 2 1 ( ) 3 ; ( ) 5 ; ( ) 10 ; ( ) 2; ( ) 4 ; ( ) 10 ; 4 - a b c d e or f 8 6 2 2 1 ( ) 3 ; ( ) 5 ; ( ) 6 6 - g h i or .
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48 REFERENCES Heinemann (2001). Edexcel GCSE Mathematics: Higher course . Oxford, Heinemann Educational Publishers.
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