Indices © T Madas

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

The Rules of Indices © T Madas

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

Rule two: a n ÷ a m = a n – m e.g. 35 ÷ 32 = 3 5 – 2 = 33 Why does it work? 35 3 x 3 x 3 x 3 x 3 35 ÷ 32 = = = 3 x 3 x 3 = 33 32 3 x 3 © T Madas

1 Rule three: a -n = a n 1 e.g. 4-3 = 43 Why does it work? 42 4 x 4 1 – 5 = 42 ÷ 45 = = = 45 4 x 4 x 4 x 4 x 4 43 © T Madas

a = 1 a special result This is true for all values of a = 1 This is true for all values of a , even if a = 0 50 = 1 0.250 = 1 (-3)0 = 1 1 2 = 1 00 = 1 © T Madas

a special result Why is it a 0 = 1? a 4 a x a x a x a a 0 = a 4 – 4 = ÷ a 4 = = = 1 a 4 a x a x a x a © T Madas

n m Rule four: a m = a m x n = a n 3 2 e.g. 72 = 7 2 x 3 = 76 = 73 Why does it work? 3 72 = 72 x 72 x 72 = ( 7 x 7 ) x ( 7 x 7 ) x ( 7 x 7 ) = 7 x 7 x 7 x 7 x 7 x 7 = 7 6 = 7 x 7 x 7 x 7 x 7 x 7 = ( 7 x 7 x 7 ) x ( 7 x 7 x 7 ) = 73 x 73 2 = 73 © T Madas

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

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

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

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

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

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

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

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

R u l e s o f I n d i c e s S u m m a r y © T Madas

R u l e s o f I n d i c e s S u m m a r y 1. a n x a m = a n + m 2. a n ÷ a m = a n – m 3. 1 Special Results a 0 = 1 a 1 = a 1n = 1 0n = 0 (unless n = 0) a -n = a n n m 4. a m = a m x n = a n 5. 1 n a = n a 6. m m n a = n a m = a n 7. ( a b ) n = a n b n 8. a b n a b n = n © T Madas

Revision on the rules of indices © T Madas

Evaluate the following, giving your final answers as simple as possible: 22 25 = 2 2 + 5 = 27 = 128 03 = x 1 2 81 = 81 = 9 77 ÷ 72 = 7 7 – 2 = 75 1 1 4 3 3 4 3 3 64 27 2-4 = = = = 24 16 3 60 = 1 15 = 1 71 = 7 3 3 2 3 16 = 2 16 = 4 = 64 2 23 = 2 2 x 3 = 26 = 64 1 = 4 2 = 16 1 3 4-2 27 = 3 27 = 3 © T Madas

Evaluate the following, giving your final answers as simple as possible: 23 23 = 2 3 + 3 = 26 = 64 06 = x 1 2 25 = 25 = 5 48 ÷ 43 = 4 8 – 3 = 45 1 1 2 3 3 2 3 3 8 27 5-2 = = = = 52 25 3 40 = 1 1-1 = 1 31 = 3 4 4 3 4 27 = 3 27 = 3 = 81 4 22 = 2 2 x 4 = 28 = 256 1 = 2 3 = 8 1 4 2-3 16 = 4 16 = 2 © T Madas

Test on the Rules of Indices © T Madas

Evaluate the following, giving your final answers as simple as possible: 22 25 = 2 2 + 5 = 27 = 128 03 = x 1 2 81 = 81 = 9 77 ÷ 72 = 7 7 – 2 = 75 1 1 4 3 3 4 3 3 64 27 2-4 = = = = 24 16 3 60 = 1 15 = 1 71 = 7 3 3 2 3 16 = 2 16 = 4 = 64 2 23 = 2 2 x 3 = 26 = 64 1 = 4 2 = 16 1 3 4-2 27 = 3 27 = 3 © T Madas

Evaluate the following, giving your final answers as simple as possible: 23 23 = 2 3 + 3 = 26 = 64 06 = x 1 2 25 = 25 = 5 48 ÷ 43 = 4 8 – 3 = 45 1 1 2 3 3 2 3 3 8 27 5-2 = = = = 52 25 3 40 = 1 1-1 = 1 31 = 3 4 4 3 4 27 = 3 27 = 3 = 81 4 22 = 2 2 x 4 = 28 = 256 1 = 2 3 = 8 1 4 2-3 16 = 4 16 = 2 © T Madas

© T Madas

Calculate the following, using the rules of indices: x3 x x4 = x7 y6 x y-4 = y2 a6 a4 = 8n6 4n4 = a2 2n2 1 w2 p0 = 1 w-2 = 4x2 x 2x3 = 8 x 5x2 x 2y3 = 10 x y 5 2 3 (x3 ) = 4 (x -2) = -3 x12 x6 4ab4 x 3a2b3 = 12 a b 4a4b2 x 5a2b3 = 20 a b 3 7 6 5 n6m3 n4m2 = n5m5 n-4m4 = n2 m n9 m © T Madas

Quick Test © T Madas

Calculate the following, using the rules of indices: x3 x x4 = x7 y6 x y-4 = y2 a6 a4 = 8n6 4n4 = a2 2n2 1 w2 p0 = 1 w-2 = 4x2 x 2x3 = 8 x 5x2 x 2y3 = 10 x y 5 2 3 (x3 ) = 4 (x -2) = -3 x12 x6 4ab4 x 3a2b3 = 12 a b 4a4b2 x 5a2b3 = 20 a b 3 7 6 5 n6m3 n4m2 = n5m5 n-4m4 = n2 m n9 m © T Madas

© T Madas

“expand” the following brackets: © T Madas

“expand” the following brackets: © T Madas

© T Madas

“expand” the following brackets: © T Madas

“expand” the following brackets: © T Madas

Just a nice puzzle on Powers Where are you going? Just a nice puzzle on Powers No way… © 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 3125 = 5 2 x 1 + 3 128 = 2 8 – 1 4096 = 4 x 9 + 6 7 + 6 + 2 216 = 6 2 + 1 32768 = 8 3 625 = 5 6 – 2 20736 = ( 6 x 2 ) 7 – 3 + © T Madas

© 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 2 x 3 x 5 = 22 x 31 x 51 30 2 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 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

© T Madas

If x = 2m and y = 2n , express the following in terms of x and/or y only: 1. 2m + n 2. 23m 3. 2n – 2 1. 2m + n = 2m x 2n = x x y = xy [ ]3 2. 23m = 23 x m = 2m = x 3 1 4 y 4 3. 2n – 2 = 2n x 2-2 = 2n x = y x 1 4 = y 4 = 2n ÷ 22 = 2n ÷ 4 = y ÷ 4 = © T Madas

© T Madas

express x in the form 5 p , where p is an integer find y find z -1 If x = 512, y = 29 x 36 and z = ⅕ : express x in the form 5 p , where p is an integer find y find z -1 1 2 1 3 1 2 x 1 2 = 512 = 5 12 1 2 = 56 x 1 3 y 1 3 1 3 1 3 = 29 x 36 = 29 x 36 = 23 x 32 = 72 x x 1 1 5 -1 1 5 1 z -1 = = = = = 5 1 5 1 5 © T Madas

© T Madas

Calculate the following: x3 x x4 = x7 a6 a4 = a2 p0 = 1 4x2 x 2x3 = 8 x 5 (x3 ) = 4 x12 4ab4 x 3a2b3 = 12 a b 3 7 n6m3 n4m2 = n2 m © T Madas

© T Madas