T Madas What is the meaning of the

What is the meaning of the words: index/indices? = power 2 6 Index Power

Rule three: a -n e. g. 4 -3 = 1 an 1 = 3

0 a =1 This is true for all values of a , even if

You better learn the last 2 rules which are very important in algebra ©

Rules n of Indices n +m m 1. a x a = a n

Revision on the rules of indices © T Madas

Evaluate the following, giving your final answers as simple as possible: 2 5 2

Evaluate the following, giving your final answers as simple as possible: 3 3 2

Calculate the following, using the rules of indices: x 3 x x 4 =

“expand” the following brackets: © T Madas

Where are you going? Just a nice puzzle on Powers No way… © T

Make the numbers in the following list by using only the digits contained in

1. Write 60 as a product of its prime factors. 2. Hence write 606

If x = 2 m and y = 2 n , express the following

Calculate the following: x 3 x x 4 = x 7 a 6 2

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What is the meaning of the words: index/indices? = power 2 6 Index Power Exponent Base © T Madas

Rule one: a n 2 x a m 4 =a e. g. 5 x 5 = 5 n +m 2+4 = 56 Why does it work? 52 x 54 = ( 5 x 5 ) x ( 5 x 5 x 5 ) = 5 x 5 x 5 x 5 = 56 Warning 52 + 54 56 © T Madas

n m 5 2 Rule two: a ÷ a =a e. g. 3 ÷ 3 = 3 n –m 5– 2 = 33 Why does it work? 5 3 3 x 3 x 3 5 2 3 ÷ 3 = 2 = = 3 x 3 = 33 3 3 x 3 © T Madas

Rule three: a -n e. g. 4 -3 = 1 an 1 = 3 4 Why does it work? 2 4 4 x 4 1 2– 5 -3 2 5 4 =4 = 4 ÷ 4 = 5= = 3 4 4 x 4 x 4 4 © T Madas

0 a =1 This is true for all values of a , even if a = 0 50 = 1 0. 250 = 1 (-3)0 = 1 1 2 0 =1 00 = 1 © T Madas

Why is it a 0 = 1? 4 a a xa xa xa 4– 4 0 4 4 a =a = a ÷a = 4 = = 1 a a xa xa xa © T Madas

Rule four: a e. g. 7 m n 2 3 = a =7 m xn 2 x 3 = a 6 n m = 7 3 2 Why does it work? 7 2 3 = 72 x 72 = (7 x 7 ) x (7 x 7 ) =7 x 7 x 7 x 7 6 =7 =7 x 7 x 7 x 7 = (7 x 7) x (7 x 7) = 73 x 73 2 3 = 7 © T Madas

1 n Rule five: a = e. g. 1 2 2 1 3 3 1 4 4 1 5 5 36 = 64 = 81 = 32 = n a 36 = 6 64 = 4 81 = 3 32 = 2 Why? © T Madas

1 n Rule five: a = e. g. 1 2 2 1 3 3 1 4 4 1 5 5 36 = 64 = 81 = 32 = n a 16 36 = 6 64 = 4 81 = 3 32 = 2 Why? © T Madas

1 n Rule five: a = e. g. 1 2 2 1 3 3 1 4 4 1 5 5 36 = 64 = 81 = 32 = n a 36 = 6 64 = 4 81 = 3 32 = 2 16 x 16 = 4 x 4 = 16 1+1 = 16 2 2 1 1 = 16 2 x 16 2 16 = 16 1 2 Why? © T Madas

1 n Rule five: a = e. g. 1 2 2 1 3 3 1 4 4 1 5 5 36 = 64 = 81 = 32 = n 3 a 36 = 6 64 = 4 81 = 3 = = = 27 x 3 27 3 x 3 27 1 + 1+ 1 27 3 3 3 1 1 1 27 3 x 27 3 32 = 2 3 Why? 27 = 16 1 3 © T Madas

m n Rule six: a = 2 3 8 = 3 3 2 2 3 5 5 3 4 4 16 = 32 = 81 = 8 2 = 3 2 3 32 = 5 3 4 81 = a m = n 64 = 4 3 16 = n a m 2 3 3 3 2 2 3 5 5 3 4 4 8 = 4096 = 64 32768 = 8 16 = 32 = 531441 = 27 81 = 2 2 8 = 2 =4 3 3 16 = 4 =64 3 3 32 = 8 3 3 81 = 3 =27 Why does this rule work? m n m x n 1 m n 1 n xm a =a =a m n 1 =a n 1 m = = n am n a m © T Madas

You better learn the last 2 rules which are very important in algebra © T Madas

n n n Rule seven: (ab ) = a b e. g. 2 2 2 (3 n ) = 3 x n = 9 n ab 23 3 6 2 3 6 =a x b = a b Why does it work? 4 (2 x 3 ) = (2 x 3 ) x (2 x 3 ) = 2 x 3 x 2 x 3 = 2 x 2 x 3 x 3 x 3 x 3 4 =2 x 3 4 © T Madas

n n a a Rule eight: = n b b e. g. n 4= n 4 4 2 16 2 π 2= π 2 3 3 2 9 Why does it work? 4 24 2 x 2 x 2 x 2 2 = = = 4 3 3 x 3 x 3 x 3 3 3 © T Madas

Rules n of Indices n +m m 1. a x a = a n –m n m 2. a ÷ a = a 1 -n 3. a = n 4. a m n 1 n = a 5. a = m n 6. a = n m xn am = n n 7. (ab ) = a b 8. = a n m a 0 = 1 a 1 = a a n n Special Results a n Summary n a m 1 n = 1 0 n = 0 (unless n = 0) a =a n b b n © T Madas

Revision on the rules of indices © T Madas

Evaluate the following, giving your final answers as simple as possible: 2 5 2 x 2 = 2 1 2 81 = 2 -4 2+5 = 27 = 128 81 = 9 1 1 = 4 = 16 2 03 = 0 7 2 7 ÷ 7 = 7 4 = 4 3 = 64 33 27 3 15 = 1 1 3 2 7 =7 2 3 2 1 3 16 = =2 27 = 3 2 x 3 = 26 = 64 27 = 3 = 75 3 0 6 =1 7– 2 2 3 3 16 = 4 = 64 1 2 = 4 = 16 -2 4 © T Madas

Evaluate the following, giving your final answers as simple as possible: 3 3 2 x 2 = 2 1 2 25 = 5 -2 3+3 = 26 = 64 25 = 5 1 1 = 25 5 4 =1 3 =3 1 4 4 ÷ 4 = 4 8– 3 = 45 3 2 = 23 = 8 33 27 = =2 16 = 3 4 3 1 2 8 1 -1 = 1 0 2 4 06 = 0 4 2 x 4 = 28 = 256 16 = 2 3 4 4 27 = 3 = 81 1 3 = 2 =8 -3 2 © T Madas

Calculate the following, using the rules of indices: x 3 x x 4 = x 7 y 6 x y-4 = y 2 a 6 2 a = a 4 p 0 = 1 8 n 6 2 2 n = 4 n 4 1 -2 w = w 2 4 x 2 x 2 x 3 = 8 x 5 5 x 2 x 2 y 3 = 10 x 2 y 3 -3 4 (x -2) = x 6 (x 3 ) = x 12 4 ab 4 x 3 a 2 b 3 3 7 b a 12 = n 6 m 3 n 2 m n 4 m 2 = 4 a 4 b 2 x 5 a 2 b 3 = 20 a 6 b 5 n 5 m 5 9 m n = n-4 m 4 © T Madas

Calculate the following, using the rules of indices: x 3 x x 4 = x 7 y 6 x y-4 = y 2 a 6 2 a = a 4 p 0 = 1 8 n 6 2 2 n = 4 n 4 1 -2 w = w 2 4 x 2 x 2 x 3 = 8 x 5 5 x 2 x 2 y 3 = 10 x 2 y 3 -3 4 (x -2) = x 6 (x 3 ) = x 12 4 ab 4 x 3 a 2 b 3 3 7 b a 12 = n 6 m 3 2 m n = n 4 m 2 4 a 4 b 2 x 5 a 2 b 3 = 20 a 6 b 5 n 5 m 5 9 m n = n-4 m 4 © T Madas

Make the numbers in the following list by using only the digits contained in each number. Each digit may only be used once. You can use any mathematical symbols and operations. 125 = 5 2+1 128 = 2 8– 1 216 = 6 2+1 625 = 5 6– 2 3125 = 5 2 x 1+3 4096 = 4 0 x 9+6 32768 = 8 7+6+2 3 20736 = (6 x 2 ) 7 – 3+0 © T Madas

1. Write 60 as a product of its prime factors. 2. Hence write 606 as a product of its prime factors 60 = 2 x 3 x 5 = 22 x 31 x 51 2 30 2 15 3 5 5 1 © T Madas

1. Write 60 as a product of its prime factors. 2. Hence write 606 as a product of its prime factors 60 = 2 x 3 x 5 = 22 x 31 x 51 (a n)m = a nm (ab )n = a n b n 606 = (22 x 3 x 5)6 = 212 x 36 x 56 © T Madas

If x = 2 m and y = 2 n , express the following in terms of x and/or y only: 1. 2 m + n 2. 23 m 3. 2 n – 2 1. 2 m + n = 2 m x 2 n = x 2. 2 3 m = 2 3 x m 3. 2 n – 2 = 2 n =[2 x x y = xy ] = x 3 m 3 2 -2 = 2 n x 1 4 =y x 1 4 = y 4 = 2 n ÷ 2 2 = 2 n ÷ 4 = y 4 © T Madas

If x = 512, y = 29 x 36 and z = ⅕ : 1. express x find y 3. find z -1 x y 1 3 in the form 5 p , where p is an integer 1 3 2. 1 2 =5 12 1 2 9 =5 = 2 x 3 z -1 = 1 5 -1 6 12 x 1 2 1 3 =5 = 2 9 x 1 3 1 1 1 = = 1 1 5 5 6 x 3 = 6 x 1 3 = 23 x 32 = 72 5 =5 1 © T Madas

Calculate the following: x 3 x x 4 = x 7 a 6 2 a = a 4 p 0 = 1 4 x 2 x 2 x 3 = 8 x 5 4 (x 3 ) = x 12 4 ab 4 x 3 a 2 b 3 3 7 b a 12 = n 6 m 3 2 m n = n 4 m 2 © T Madas