1. 25 an = 記住:a zero Index Concepts 2  2  2  2  2 a  a  a  …  a
(5 times)
index/exponent
an =
(n times)
a  a  a  …  a
base
“a to the power n”
記住:a zero

2. Law of positive integral indices
Class Activity 1
23 x 24 = (2 x 2 x 2) x (2 x 2 x 2 x 2) = 23+4 = 27
a2 x a3 = (a x a) x (a x a x a) = a2+3 = a5
b5 x b4 = (b x b x b x b x b) x (b x b x b x b) = b5+4 = b9
am  an = 
a  a  …  a
(m times)
(n times)
a  a  …  a
(m x n times) =
am  an
= am + n
am  an
law 1

3. Class Activity 2

4. =
, where m > n and a  0
, where m < n and a  0
Law 2

5. (a) a7.a3 = a7+3 = a10 an.a3 = an+3 (b) a7 a9-2 = (c) (d)
EXAMPLE 1 – Simplify the following expressions
(a)
a7.a3 =
a7+3 =
a10
an.a3 =
an+3
(b) a7
a9-2 =
(c)
(d)

6. a (23)2 = (2  2  2)  (2  2  2) = 232= 26 (a4)3 =
Class Activity
(23)2 =
(2  2  2)  (2  2  2) =
232= 26
(a4)3 =
(a  a  a  a) 
(a  a  a  a) =
a43= a12
(b3)5=
(bbb) 
b35= a15
(bbb)=
n times of am
(aa…a) 
(aa…a) …… 
nm times of a
(aa…a) m n a
Law 3

7. (a5)2= a52= a10 (an)3= an3= a3n (a4)m= a4m= a4m (a) (b) (c) x12
Example 2- Simplify the following expressions
(a5)2=
a52=
a10
(a)
(an)3=
(b)
an3=
a3n
(a4)m=
a4m=
(c)
a4m
Classwork- Simplify the following expressions
x12
(x4)3=
x43=
(1)
x6n
(xn)6=
xn6=
(2)
x5m
(x5)m=
x5m=
(3)

8. (2  3) (2  3) (2  3) (2  3) (2  3)4= =(2  2  2  2)
Class Activity
(2  3)
(2  3)
(2  3)
(2  3)
(2  3)4=
=(2  2  2  2)
(3  3  3  3)
=24 34
(3  5)5=
(3  5)
(3  5)
(3  5)
(3  5)
(3  5)
(5  5  5  5  5)
=(3  3  3  3  3)
=35 55
(ab)3=
(a  b)
(a  b)
(a  b)
=(a  a  a)
(b  b  b)
=a3 b3

9. (a  b) (a  b)…. (a  b) (ab)m= (ab)m= (a  a  …  a) 
m times of ab
(ab)m=
(a  a  …  a) 
(b  b  …  b)
m times of a
m times of b
(ab)m=
am  bn
Law 4

10. 4 4 = = = 4 5 5 = = = 5 3 3 = = = 3
(n times of )
Law 5
(where b  0)

11. EXAMPLE 3: Simplify the following expressions4 4 4 =
(b) = 4 2 2 3 3 2 3 6
(c) 3 = = = 3 3 3
Classwork 1.3 3
(2) =
(2x)5=
25x5=
32x5
(1) 2
(3) = 5 2

12. EXAMPLE 4: Simplify the following expressions
(b)
(c)

13. Classwork 1.4 1. 2. 40 3.

14. (1b) a2.a3.a4=a2+3+4=a9 (2b) (3b) (x7)3 =x73 =x21 (4b)
Exercise 1A
Level 1
(1b)
a2.a3.a4=a2+3+4=a9
(2b)
(3b)
(x7)3 =x73 =x21
(4b)
(4a2)3 =43. (a2)3=64. a23 =64. a6

15. (5b) 23.(b2)3. 42(b3)2 (6b) (2b2)3(4b3)2= =23.(b2)3. 42(b3)2
Exercise 1A
Level 2
(5b)
23.(b2)3. 42(b3)2
(6b)
(2b2)3(4b3)2=
=23.(b2)3. 42(b3)2
=8. b6. 16b6
=8. 16 b6. b6
=128 b6+6
=128 b12

16. (7b)
(8b)

17. (9b)
(10b)

18. Mistakes students always make
23  54=
(2  5) 7
(2  5) 7=
10 7
Wrong !
since these numbers has no common base.
3  46=
126
Wrong !
since these numbers has no common base.
(x3)2=
x3+2= x5
3a.2a=5a

19. 1 ∵ 
Thus,
(where a  0)
Law 6

20. From Law 2
Any integer m and n ∴
Thus
Law 7

21. Example 5- Evaluate the following expressions= 3 = =
(a)
(-2)-3 = 2 = -2
(b) = = 3 2
(1) 2
(3-3)2(20)
(c) = = = 3 2

22. Classwork 1.5 1. 2. 3. 4.

23. = c b a = ) c b a ( = ) ( c b a 2 q p = q p = 2 q p a c b = . q p (a)
Example 6 - Simplify the following expressions and express your answers in positive indices
(a) = - 3 1 2 . q p
(b) 2 3 q p = - ) 3 ( 1 2 q p = - 3 1 2 q p 4 2 a c b = - 2 4 c b a = - 2 1 ) c b a = - 2 1 ) ( c b a
(c) (

24. Classwork 1.6
(1)
(2) 4 3 2 1 27 . ) ( x x1 = - +
(3)
(4)

25. [ ] [ ] ) ( b a = ) )( ( b a ) )( ( b a = b a = b a = b a = ) ( y x =
Example 7: Simplify the following expressions and express your answers in positive indices
(a) 2 1 3 ) ( b a = - 4 2 6 ) )( ( b a - 4 6 2 ) )( ( b a = - 10 6 b a = 4 6 2 b a = - + 10 6 b a = - [ ] 1 3 2 ) ( y x = - = [ ] 1 3 6 2 y x - 6 2 x y 3
(b) 3 6 1 2 y x = -

26. y x ) ( = y x = y x y x = y x = y x = ) ( y x xy = 3 . y x 3 y x = 3 .
Classwork 1.7
(1) y x 3 1 2 ) ( = - y x 3 6 4 = - y x 3 4 6 - y x 12 = y x 3 4 6 = + - y x 12 = - 1
(2) 2 3 1 ) ( y x xy = - 6 2 1 3 . y x - 6 2 1 3 y x = - 2 6 3 . y x = - + -1 4 18 xy = - 4 18 y x =

27. [ ] [ ] ) ( y x = y x = y x y x = 8 . 2 b a = 32 b a ) 8 ( 2 b a = 8 2- [ ] 2 6 y x = - 4 12 y x 4 12 y x = - 4 8 . 2 5 b a = + - 32 3 2 b a ) 8 ( 2 1 5 3 b a = - 8 2 5 1 b a = -

28. . y x = y x = y x ) ( y x = ) )( ( y x = ] ) ( 5 [ y x = ) ( 5 1 y x )
Additional example 2 8 4 . y x = 2 8 4 y x = + 10 8 y x 2 1 4 ) ( y x = - 2 4 8 ) )( ( y x = - 2 3 1 ] ) ( 5 [ y x = - 4 3 2 ) ( 5 1 y x - 4 3 2 ) ( 25 y x = 12 2 4 16 25 y x = 2 12 4 25 y x = - 10 4 25 y x =

29. 8 a )( b = . 32 . 16 ) ( y x = - Exercise 1B (1c) (1b) (3b) (4b) (5a)+
(5a)
(6a) 2 6 4 1 32 . 16 ) ( y x = - +

30. = a a a = 1 a a2x=ax 2x-x=0 (a0, a1, a-1) x=0 2x - x
1.4 SIMPLE EXPONENTIAL EQUATIONS
Equation involves unknowns in the exponents
(a0, a1, a-1)
a2x=ax a 2x = 1 a x = a 2x - x a
2x-x=0 ∴
x=0

31. (a) 6x=1 (b) 2x=8 2x=23 6x=60 ∴ x=3 ∴ x=0 2.13x=2 13x=1 x=0 ∴ 13x=130
Example 8: Solve the following exponential equations
(a)
6x=1
(b)
2x=8
2x=23
6x=60 ∴
x=3 ∴
x=0
Classwork 1.8
(2)
(4)
2.13x=2
13x=1
x=0 ∴
13x=130
(3)
4x=256
4x=44
x=4 ∴
(1)
11x=1
11x=110
x=0 ∴
3x=81
3x=34
x=4 ∴

32. (b) 42x+1=8x-2 (a) 52x=625 (22)2x+1=(23 )x-2 52x=54 22(2x+1)=23(x-2)
Example 9: Solve the following exponential equations
(b)
42x+1=8x-2
(a)
52x=625
(22)2x+1=(23 )x-2
52x=54
22(2x+1)=23(x-2)
244x+2=23x-6
2x=4
4x+2=3x-6 ∴
x=2
4x-3x=-6-2 ∴
x=-8
Classwork 1.9
(4)
94x-1=272x+4
(3)
82x=16x+4
(32)4x-1=(33) 2x+4
(1)
32x=81
(2)
42x=256
(23)2x=(24) x+4
32(4x-1)=33(2x+4)
32x=34
42x=44
26x=24(x+4)
38x-2=26x+12
2x=4
26x=24x+16
2x=4
6x=4x+16
8x-2=6x+12
x=2
6x-4x=16 ∴
8x-6x=12+2
x=2 ∴
2x=16
2x=14 ∴
x=8 ∴
x=7

33. Example 10: Solve the following exponential equations∴

34. Classwork 1.10 - Solve the following exponential equations
(1)
(1) ∴ ∴

35. Scientific Notation科學記數法
In scientific world, we often study objects which are very large or very small
e.g.
Mass of an element of hydrogen is kg.
Velocity of light = m/s
Difficult to read and write!!!
To overcome, we use a method called scientific notation to express these numbers approximately
= 1.68  corr. to 3 sig fig
= 3  108 kg
= 2.59  corr. to 3 sig fig

36. Positive number N can be expressed as N = d  10n, where n is an integer and 1  d < 10.
Example 11- express the following numbers in scientific notation
(a) = 1  = 1 10-6
(b) = 1.8  = 1 104
(c) = 2  = 2 106
(d) = 2.16  = 2.16 10-4
(e) 345  104 = 3.45  100  104 = 3.45  102  104 = 3.45  106
(f)  10-3 = 5.02  0.01  10-3 = 5.02  10-2  10-3 = 5.02  10-5

37. Classwork 1.1- express the following numbers in scientific notation
(1) = 3.74  = 1 106
(2) = 7.89  = 1 10-7
(3) 6 31  1011 = 6.311001011 = 6.31  1021011 = 6.311013
(4) 10-5=1.20370.110-5= 10-110-5 = 10-5
(5) – 103=-4.5080.0001104=-4.50810-4104=-4.50810-8
(6) 10-4 =7.0080.110-4=7.00810-110-3=7.00810-4

38. Example 12-. Evaluate the following,express your answers in scientific
Example 12- Evaluate the following,express your answers in scientific notation and correct to 3 significant figures
(a)
(3.52108)(7.911012)
=(3.527.91 )(108 1012)
= 1020
= 1011020
= 1021
=2.781021
Corr. to 3 sig. Fig.
(b) 6 3 9 10 42 . 1
41913 75 5 16 8 - = 
Corr. to 3 sig. Fig.

39. Classwork 1.12 ,express your answers in scientific notation and correct to 3 significant figures
(1)
(2.305107)(8.21012)
=18.9011012
=1.891011012
=(2.3058.2)(107 105)
=1.891013
(2)
(210-12)(71023)
=141011
=1.41011011
=(27)(10-121023)
= 1.41012
(3)
(3.810-10)(810-11)
=30.410-21
=3.0410110-21
=(3.88)(10-1010-11)
= 3.0410-20
(4) 2 1 3 10 69 . 4
46875 6 - =  10 4 6 29 . 2
2882 5 3
009 8 =  -
(5) 1 4 5 10 34 . 3
3357 67 - =
(6) 

40. (2.510-3)(3.610-4)=2.53.610(-3)+(-4) =910-7 =90.0000001
Example 13 – Express the values of the following as integers or decimal numbers
(a)
2105=2  =
(b)
3.1410-4=3.14  =
(c)
(2.510-3)(3.610-4)=2.53.610(-3)+(-4) =910-7
=9 =
Classwork 1.13
(1)
6.36106=6.36  =
(2)
4.1210-5=4.12 =
(3)
(1.310-5)(5.7107)= 1.35.7 =7.41102
=7.41100
=741 3 . 1 10 5 7 75 9 2 =  -
(4)

41. Example 14- express the following numbers in scientific notation and arrange them in ascending order
89, 98, 710, 107
Use calculator
710
107 98 89 = = = =
=1.34108
=4.30107
=2.82108
=1107 8 Xy 9 = 7 10
107< 98 < 89 < 710
Compare powers of 10. The greater the power of 10, the greater the number. If powers of 10 are equal, compare values of d of the numbers

42. Numeral Systems Denary System (十進制)- we use everyday
In the algebraic expression ax3+ bx2 + cx1 + dx0
x is an integer more than 1.
a, b, c and d are all non-negative integers less than x.
Denary System (十進制)- we use everyday
When x = 10, abcd is a denary number.
abcd = a  b  c  d  100
e.g (10)=3     100
3065(10)=3     1
The place value of each digit is ten times the place value of the digit on its right-hand side.
Place values
ten basic numerals: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 (less than 10)
Place holder
Distinguish values between 365 and 3065

43. Example 15
(a) Write down the place values of each digit of 50943
Digit 5 9 4 3
Place Value
10000
1000
100 10 1
(b) Express in the expanded form with base 10
50943(10)=5     1
=5    100

44. (2) Express the following as denary numbers
Classwork 1.15
(1) Write down the place value of each digit in the following numbers
(a) 24071
(b) 9145 1 10
100
1000
10000
Place Value 7 4 2
Digit 5 9
(2) Express the following as denary numbers
(a) 2541 = 2   100
(b) = 2     100

45. What are the expansions of 12.302?
Challenge
What are the expansions of ?
= 1      10-3
Example 16 Express the following as denary numbers.
(a)
2    100 = =237
(b)
9     100
=9       100
=905067

46. Classwork 1.16 Express the following as denary numbers.
(1)
6 101
=6   100
= 6030
(2)
4 10 + 9
=4      100
=40029
5    10
=5      100
=50430
(3)

47. Exercise 1E
(1) Write down the place value of each digit 2 in the following denary numbers
(a) 1232
100, 1
(b)
10000, 10
(2) Express the following denary numbers in the expanded form with base 10
(a) 119= 1    100
(b) 2318= 2     100
(3) Express the following as denary numbers
(a) 6  1 = 6   100 =68
(c) 8   = 8    101+ 1100
=800401
(e) 1   1 = 1  101+ 9100 =1409
(4) Find the greatest and smallest denary numbers
(a) 2, 6, 4
Greatest 642, smallest 246
Greatest 5310, smallest 1035
(b) 0,1,3,5
(5) (a) 40014
10000, 1
10000/1=10000
10000, 10
10000/10=1000
(b)

48. Denary System Binary System 1 1 1 1 =10111(2)
$100
213(10) = 2    1
$10 $1
0 to 9 (<10)
213(10) = 2    100
Binary System
23(10)=      1 1 1 1 1
0 to1 (<2)
=10111(2)
10111(2)= 1     20

49. Denary System Binary System 1 1 1 1 =10111(2)
$100
213(10) = 2    1
$10 $1
0 to 9 (<10)
213(10) = 2    100
Binary System 1
$16 $8 1 $4 1 $2 1 $1
23(10)=     
0 to1 (<2)
=10111(2)
10111(2)= 1     20

50. Binary System (二進制)- only computer understand
Denary System (十進制)
e.g (10)=3     100
0 to 9 (<10)
Binary System (二進制)- only computer understand
When x = 2, abcd is a binary number.
abcd(2)= a  b  c  d  20
e.g (2)=1     20
1011(2)=1     1
The place value of each digit is two times the place value of the digit on its right-hand side.
Place values
2 basic numerals: 0 to1 (< 2)
Place holder
Distinguish values between 111 and 1011

51. =22=4 =24=16 Digit 1 Place Value 32 16 8 4 2 1 2 4 8 16 Place Value
Example 17
(a) Write down the place value of each digit in (2)
Digit 1
Place Value 32 16 8 4 2
(b) Find the place value of the digit 0 in (2)
=22=4
Classwork 1.17
(a) Write down the place value of each digit in (2) 1 2 4 8 16
Place Value
Digit
=24=16
(b) Find the place value of the digit 0 in (2)

52. Example 18
Express the following binary numbers in the expanded form with base 2
(a) 1101(2)=1     20
(b) (2)=1   23 + 022 + 121 + 120
Classwork 1.18
Express the following binary numbers in the expanded form with base 2.
(1) (2)=1      20
(2) (2) =1  23 + 122 + 121 + 020

53. Example 19 Express the following as binary numbers.
1     1
=1     20
=1100(2)
(b)
=1       20
1    1
=101001(2)
(2)
Classwork 1.19
(1)
1   4
=1      1
=10100(2)
=1      20
1  
(2)
=1      1
=10101(2)

54. Conversion of Binary number into Denary number
Example 20
(b) Convert (2) into a denary number
101010(2)
=1       20 =
=42
Classwork 1.20
(b) Convert (2) into a denary number
=1       20
101111(2) = =
(1)
=47
(2)
(b) Convert (2) into a denary number
(2)
=1     22+ 121 + 020 =
=242

55. Example 21 Convert 10011(2) into a denary number
10011 (2) =
1      20
=1      1 =
=19
Classwork 1.21
(b) Convert (2) into a denary number
=1       20
111000(2) =
=32+8+4
(1)
=56
(2)
(b) Convert (2) into a denary number
(2)
= 1    22+ 121 + 020 =
=86

56. Conversion of Denary number into Binary number
Method 1- Fill in the number from left to right
30(10) = 1 1 1 1
= 11110(2)

57. Conversion of Denary number into Binary number
Method 2- divide the denary number by 2 successively
write the answer in terms of remainders from bottom to top 2) 30 2) 15 2) 7 1 2) 3 1 2) 1 1 1
11110(2)
30(10) =

58. Example 22- Convert the following denary into binary number2) 32 8 16 4 2 1
(a) 21 2) 21 5 10 2 1
(b) 32
=10101(2)
=100000(2)

59. Classwork 1.22- Convert the following denary into binary number
=11100(2) 2) 28 7 14 3 1
(1) 11 2) 11 2 5 1
=1011(2)
(2) 28 2) 64 16 32 8 4 1 2
= (2)
(3) 64

60. Hexadecimal System (十六進制)
Binary System
0,1 (<2)
23(10)=      1
$16 1 $8 $4 1 1 $2 $1
=10111(2)
12F(10)
10111(2)= 1     20
Hexadecimal System (十六進制)
=303(10) = 1   F 
$256
$16 $1
=12F(16)
0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F (<16)
12F(16)=1   F  160

61. Digit 3 9 A 1 Place Value 4096 163 256 162 16 161 160 Digit 9 F C
Example 23
(a) Write down the place value of each digit in 39A1(16)
Digit 3 9 A 1
Place Value
4096
163
256
162 16
161
160
(b) Find the place value of the digit in 4A067(16)=162=256
Classwork 1.23
(a) Write down the place value of each digit in 9FFC(16)
Digit 9 F C
Place Value
4096
256 16 1
(a) Write down the place value of digit A in FC6A7(16) =16

62. Example 24 (a) 82A(16)=8  162 + 2  161 + A  160
Express the following hexadecimal numbers in the expanded form with base 16
(a) 82A(16)=8   A  160
(b) ABCD(16)=10     160
Classwork 1.24
Express the following hexadecimal numbers in the expanded form with base 16
(1) F3CF(16)=15     160
(2) 100FFF(16)=1     160

63. Example 25
Convert the following hexadecimal numbers into denary numbers
(a) 123(16)=1    160= = 291(10)
(b) ABC (16)=10    160
=10  
=2748
Classwork 1.25
(a) 2F3(16)=2   160= = 755(10)
(b) 3DD (16)=3  160 = =989(10)
(c) AFFF(16)=10   160 = =989(10)

64. Example 26 – Convert the following denary numbers into hexadecimal numbers
=159(16)
16)
16) 45
345
16) 9
16) 2
13 (D) 21
16) 5 2 1 1

65. Classwork 1.26 – Convert the following denary numbers into hexadecimal numbers
(1) 321(10)
=141(16)
(2) 405(10)
=195(16)
16)
321
16)
405
16) 1
16) 5 20 25
16) 4 1
16) 9 1 1 1
(3) 4110(10)
=100E(16)
16)
4110
16)
14(E)
256
16) 16
16) 1 1