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Indices and standard form

# Indices and standard form - MEP Y9 Practice Book A 3...

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37 3 Indices and Standard Form 3.1 Index Notation Here we revise the use of index notation. You will already be familiar with the notation for squares and cubes a a a a a a a 2 3 = × = × × , and this is generalised by defining: a a a a n = × × × ... 1 2 44 3 44 n of these Example 1 Calculate the value of: (a) 5 2 (b) 2 5 (c) 3 3 (d) 10 4 Solution (a) 5 2 = 5 5 × = 25 (b) 2 5 = 2 2 2 2 2 × × × × = 32 (c) 3 3 = 3 3 3 × × = 27 (d) 10 4 = 10 10 10 10 × × × = 10 000 Example 2 Copy each of the following statements and fill in the missing number or numbers:

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MEP Y9 Practice Book A 38 Solution (a) 2 7 = 2 2 2 2 2 2 2 × × × × × × (b) 9 = 3 3 3 2 × = (c) 1000 = 10 10 10 10 3 × × = (d) 5 3 = 5 5 5 × × Example 3 (a) Determine 2 5 . (b) Determine 2 3 . (c) Determine 2 2 5 3 ÷ . (d) Express your answer to (c) in index notation. Solution (a) 2 5 = 32 (b) 2 3 = 8 (c) 2 2 5 3 ÷ = 32 8 ÷ = 4 (d) 4 = 2 2 Exercises 1. Calculate: (a) 2 3 (b) 10 2 (c) 3 2 (d) 10 3 (e) 9 2 (f) 3 3 (g) 2 4 (h) 3 4 (i) 7 2 2. Copy each of the following statements and fill in the missing numbers: (a) 10 10 10 10 10 10 × × × × = (b) 3 3 3 3 3 × × × = 3.1
MEP Y9 Practice Book A 39 (c) 7 7 7 7 7 7 × × × × = (d) 8 8 8 8 8 8 × × × × = (e) 5 5 5 × = (f) 19 19 19 19 19 × × × = (g) 6 6 6 6 6 6 6 6 × × × × × × = (h) 11 11 11 11 11 11 11 × × × × × = 3. Copy each of the following statements and fill in the missing numbers: (a) 8 2 = (b) 81 3 = (c) 100 = 10 (d) 81 = 9 (e) 125 = 5 (f) 1 000 000 = 10 (g) 216 = 6 (h) 625 = 5 4. Is 10 2 bigger than 2 10 ? 5. Is 3 4 bigger than 4 3 ? 6. Is 5 2 bigger than 2 5 ? 7. Copy each of the following statements and fill in the missing numbers: 8. Calculate:

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MEP Y9 Practice Book A 40 9. Calculate: (a) 3 2 4 + ( 29 (b) 3 2 4 - ( 29 (c) 7 4 3 - ( 29 (d) 7 4 3 + ( 29 10. Writing your answers in index form, calculate: 11. (a) Without using a calculator, write down the values of k and m. 64 8 4 2 2 = = = k m (b) Complete the following: 2 15 = 32 768 2 14 = (KS3/99Ma/Tier 5-7/P1) 3.2 Laws of Indices There are three rules that should be used when working with indices: When m and n are positive integers, 1. a a a m n m n × = + 2. a a a m n m n ÷ = - or a a a m n m n = - m n ( 29 3. a a m n m n ( 29 = × These three results are logical consequences of the definition of a n , but really need a formal proof. You can 'verify' them with particular examples as below, but this is not a proof: 2 2 7 3 × = 2 2 2 2 2 2 2 2 2 2 × × × × × × ( 29 × × × ( 29 = 2 2 2 2 2 2 2 2 2 2 × × × × × × × × × = 2 10 (here m n m n = = + = 7 3 10 , and ) 3.1
MEP Y9 Practice Book A 41 or, 2 2 7 3 ÷ = 2 2 2 2 2 2 2 2 2 2 × × × × × × × × = 2 2 2 2 × × × = 2 4 (again m n m n = = - = 7 3 4 , and ) Also, 2 7 3 ( 29 = 2 2 2 7 7 7 × × = 2 21 (using rule 1) (again m n m n = = × = 7 3 21 , and ) The proof of the first rule is given below: Proof a a m n × = a a a × × × ...

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