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3. 2323 2424 2020 3030 2 -1 3 -1 4 -2 2 5  2 2 2 6  2 2 2 3  2 3 3 6  3 6 2 3  2 4 3 5  3 6 4 7  4 9 Write the following as a single exponent and evaluate 816111/21/31/16 Write the following fractions in index form. Write the following as fractional powers. a m x a n = a m+n Multiplication Rule a m  a n = a m-n Division Rule a 0 = 1 Negative Index Rulea -n = 1/a n

4. The Rules for IndicesDivision Consider the following: a m  a n = a m-n Division Rule Generalising gives: Using this convention for indices means that: For division of numbers in the same base you?subtract the indices a 0 = 1 In general: and Generalising gives: Negative Index Rule

5. a m x a n = a m+n Consider the following: 32 32 x 33 33 = 3 x 3 x 3 x 3 x 3 = 3 5 (base 3) 24 24 x 23 23 = 2 x 2 x 2 x 2 x 2 x 2 x 2 = 2 7 (base 2) 53 53 x 52 52 x 5 = 5 x 5 x 5 x 5 x 5 x 5 = 5 6 (base 5) For multiplication of numbers in the same base you? Multiplication Rule 3434 base 3 index 4 5353 base 5 index 3 add the indices Generalising gives: 2828 3737 4 10 54546 8 12 2929 2 3 x 2 5 3 2 x 3 5 4 6 x 4 4 5 3 x 5 1 6 3 x 6 3 8 3 x 8 9 2 7 x 2 2 Write the following as a single exponent: The Rules for Indices:Multiplication

6. 2 -1 3 -2 4 -2 5 -3 6 -2 8 -2 2 -4 2 2 x 2 -3 3 4 x 3 -6 4 -4 x 4 2 5 2  5 5 6 3  6 5 8 7  8 9 2 4  2 8 Write the following as a single exponent and evaluate: 2 -3 x 2 -2 Write the following as a single exponent and evaluate: 3 -1 x 3 -2 4 -4 x 4 3 2 -3  2 2 7 -1  7 -1 4 3  4 -1 2 -5 3 -3 4 -1 2 -5 7 0 = 14 4 = 256

7. The Rules for Indices:Powers Consider the following: (3 2 ) 3 = 3 x 3 x 3 x 3 x 3 x 3 = 3 6 (base 3) (2 4 ) 2 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 = 2 8 (base 2) (5 3 ) 3 = 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 = 5 9 (base 5) To raise an indexed number to a given power you?multiply the indices (a m ) n = a mn Power Rule Generalising gives: 2626 3434 4 12 5656 6 -6 8 -4 2 -14 (2 2 ) 3 (3 2 ) 2 (4 3 ) 4 (5 3 ) 2 (6 -3 ) 2 (8 -2 ) 2 (2 7 ) -2 Write the following as a single exponent:

8. y5y5 6y 6 30p 6 48k a5b9a5b9 12a 6 b 8 = 2pq 2 x 2pq 2 = 4p 2 q 4 = 3a 2 b 3 x 3a 2 b 3 = 9a 4 b 6 = 5m 2 n 3 x 5m 2 n 3 = 25m 4 n 6 = 8p 3 q 6 Raise the number to the given power and multiply the indices. = 81a 8 b 12 = 32m 10 n 15 = 2pq 2 x 2pq 2 x 2pq 2 Indices in Expressions Simplify each of the following: y 2 x y 3 2y 2 x 3y 4 5p 2 x 3p 3 x 2p 8k 3 x 2k -4 x 3k 2 ab 2 x a 2 b 3 x a 2 b 4 2a 3 b 2 x 3ab 4 x 2a 2 b 2 (2pq 2 ) 2 (3a 2 b 3 ) 2 (5m 2 n 3 ) 2 1 2 3 4 5 6 7 8 9 (2pq 2 ) 3 (3a 2 b 3 ) 4 (2m 2 n 3 ) 5 10 11 12

9. 1 2 3 4 5 6 Simplify the following: 1 5 1 3 1 3 2 1 4 4 2 3 1 3 2 32 3 44 1 1 1 1

10. Write the following as a power of 2 Write the following as a power of 3 Write the following as a power of 5