1. Indices
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2. What is the meaning of the words:
index/indices?
= power
Index
Power
Exponent 6 2
Base
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3. The Rules of Indices
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4. Rule one: a n a m = a n + m e.g. 52 54 = 5 2 + 4 = 56x
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
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5. 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
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6. 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
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7. 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
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8. 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
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9. 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
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10. Why? Rule five: a = a e.g. 36 = 36 = 6 64 = 64 = 4 81 = 81 = 3 32 = 32n
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?
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11. 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?
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12. 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?
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13. Why? Rule five: a = a 27 x 27 x 27 = 3 x 3 x 3 e.g. 36 = 36 = 6 = 271 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?
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14. m Rule six: a = a m = a 2 2 8 = 8 2 = 64 = 4 8 = 8 = 2 = 4 3 3 16 = 16n
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
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15. You better learn the last 2 rules which are very important in algebra
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16. Why does it work? Rule seven: ( a b ) n = a n b n e.g. ( 3 n ) 2 = 3 2x 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
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17. π π π Why does it work? Rule eight: a b a b = n 2 n 2 n 16 e.g. = = 34 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
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18. R u l e s o f I n d i c e s S u m m a r y
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19. R u l e s o f I n d i c e s S u m m a r y1.
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
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20. Revision on the rules of indices
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21. 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
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22. 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
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23. Test on the Rules of Indices
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24. 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
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25. 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
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26. © T Madas

27. 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
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28. Quick Test
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29. 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
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30. © T Madas

31. “expand” the following brackets:
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32. “expand” the following brackets:
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33. © T Madas

34. “expand” the following brackets:
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35. “expand” the following brackets:
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36. Just a nice puzzle on Powers
Where are you going?
Just a nice puzzle on Powers
No way…
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37. 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 +
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38. © T Madas

39. 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
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40. 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
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41. © T Madas

42. If x = 2m and y = 2n , express the following in terms of x and/or y only:
1. 2m + n m
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 =
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43. © T Madas

44. 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
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45. © T Madas

46. 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
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47. © T Madas