F.3 Mathematics Supplementary Notes
Chapter 2 Laws of Indices
Chapter 2 Laws of Indices
Important Terms
index notation
base
rounding off
9/2005
P. 1
Name:_________( ) Class: F.3 (
指數記數法
底數
捨入
integral index
scientific notation
整數指數
科學記數法
Revision Notes
1. Zero index and negative integral indices
When a  0 and n is a positive integer, we have:
(a) a 0  1
e.g.
(2 x 0 ) 3  (2  1) 3  8
(b) a  n 
1
an
e.g. 3  2 
1 1

32 9
e.g. 2a 2 b 
 y 
e.g.  2 
x 
1
1

 32
2
1
3
32
4
4 2 a 2b ac bc
 2

c 1
a 16 8a
 x2
 
 y
4

x8
  4
y

2. Laws of integral indices
When m and n are integers and a, b  0 , then
(a)
am  an  amn
e.g. 42 × 45 = 42 + 5 = 47 = 16384
(b)
am
 amn
n
a
e.g.
(c)
(a m ) n  a m  n
e.g. (2 3 ) 5  2 3 5  215 = 32768
(d)
(ab) m  a m b m
e.g.
(e)
am
a
   m
b
b
m
37
 37  2  35 = 243
32
(2x)4 = 24 x4 = 16x4
4
x4 x4
x
e.g.    4 
3
81
 3
3. Scientific Notation
(a)
When a positive number is expressed as a  10 n , where 1  a  10 and n is an integer, we say
that the number is expressed in scientific notation.
)
F.3 Mathematics Supplementary Notes
(b)
Chapter 2 Laws of Indices
9/2005
P. 2
The method of expressing a number x in scientific notation:
(i)
x  10 : Move the decimal point n places to the left until we get a ( 1  a  10 ).
Then x = a  10 n .
e.g. 87 395 = 8.739 5  10 4
 move the decimal point 4 places to the left
(ii) 0  x  1 : Move the decimal point n places to the right until we get a ( 1  a  10 ).
Then x = a  10  n .
e.g. 0.00 314 = 3.14  10 3
move the decimal point 3 places to the right
(iii) 1  x  10 : Multiply the number by 10 0 . Then x = x  10 0 .
e.g. 3.5 = 3.5  10 0 .
(iv) x  0 : Express the number which has the negative sign removed in scientific notation, then
add back the negative sign.
e.g. –102.534=  1.02534  10 2 .
(c) For an approximate value expressed in scientific notation, we can definitely determine its level of
accuracy.
(d) For numbers of many digits, we can use scientific notation not only to simplify the operations
involving them, but also to compare them easily.
Exercise
Level I
1. Simplify the following expressions.
(a)
a4 × a5 =
(c)
(a6)4 =
(b) c24 ÷ c3 =
(d) a (ab)2 =
3
(e)
2 3
(2c ) =
(f)
 3a 
(2a )  =
 b 
F.3 Mathematics Supplementary Notes
Chapter 2 Laws of Indices
9/2005
2
(g)
 3 
u 
 =
u
3
(h) (–a) 2 × 2a3 × 5a =
2. Find the value of each of the following expressions without using a calculator.
Give the answers in integers or fractions.
(a)  6 2  (3) 5  (8 4 )
(b)  2 2  (6) 2  (3) 3
(c)
30  3 2
(e)
(2)3
(g)
7
 
2
×
2
(d) 2 2  4 3
(32)
(f)
 1
 
 3
4
3
 1
(h)     (8)  2
 2
P. 3
F.3 Mathematics Supplementary Notes
Chapter 2 Laws of Indices
9/2005
3. Simplify the following and give the answers with positive indices.
(a) (c 0 ) 2
(b) a 2  a 3  a 5
(c)
(a 3b 2 ) 2
(d) (3a 2 )( 2b) 3
(e)
(2b 1 ) 2
(f)  a    a 6  a 6
 a 0 b 2
(g) 
 3
4.



1
6
 a 2 b 2 
(h)  2 3 
a b 
4
Express each of the following numbers in scientific notation.
(a) 80 000 000
(b) 163.102
(c) 0.001 367
(d) 0.000 060 135
P. 4
F.3 Mathematics Supplementary Notes
(e) 0.123 ×105
Chapter 2 Laws of Indices
(f)
9/2005
12.34 ×107
5. Express the following numbers as whole numbers or decimal numbers.
(a) 6 ×105
(b) 1.5 ×104
(c) 5.034 ×103
(d) 3.052 ×105
6.
Evaluate each of the following expressions without using a calculator, and express the answer in
scientific notation.
(a) (2 ×1012) ×(7 ×1023)
(b) (1.26 × 1013) ÷ (3.0 × 107)
7.
Solve the following equations:
(a) 2 x =
1
16
(c) 9 x = 2432
(b) 81 x = 27
(d) 3 x × 3 2x = 1
P. 5
F.3 Mathematics Supplementary Notes
Chapter 2 Laws of Indices
9/2005
Level II
8. Simplify the following expressions.
2
(a)
2
2
0
   (2)  (9)
3
 
(c)

(e)
 x 5   x2 2
 3
(b) 1  (2)    
 2
2
3
1  2
     (3 2 ) 3
2  3
 2a 0
(g)    2
 b




(d) 9c 2   6c 3
(f)
(  a 2 b) 3
(ab) 2
(h)
3x  22x
6x
4

3
P. 6
F.3 Mathematics Supplementary Notes
Chapter 2 Laws of Indices
n 1
(i)
9.
27
3 2 n 1
9/2005
x 2
4 8
2 x 6  4 2 x
x
(j)
Use a calculator to evaluate each of the following expressions Express the answer in scientific
notation and round off the answer to 3 significant figures if necessary.
(a) 0.000 000 000 12 ÷ 0.000 000 003
16
(c) 15
(b)
(d)
(2  10 3 ) 4
40000000000
3.3345  1013
5149900  3.66
10. Solve the following equations.
(a) 3 x × 3 2x = 1
(c)
3 x  3 x 1  6
 
(b) 24  2 x
5
3
P. 7
F.3 Mathematics Supplementary Notes
(d)
4
3x2
 
1 x
 256 8
Chapter 2 Laws of Indices
(e)

39
9/2005
x 3

 1 
 
 27 
P. 8
1 4 x
11. If 2 3 x 3  8 , find the values of
x
(a) 2
1
(b)  
8
x
12. With the current technology of computer industry, we can cut out very fine circuits on computer chips.
The width of the lines of the circuit is only 25 over 100 thousand mm. Write this quantity in scientific
notation.
13. Suppose Mr. Leung's heart beats 72 times per minute and his life-span is 90 years. How many tims
will his heart beat in his whole life?
F.3 Mathematics Supplementary Notes
Chapter 2 Laws of Indices
9/2005
P. 9
14. The distance travelled by light in one year is 9.46  1012 km. The distance between the star X and the
Earth is 5.913  1015 km.
(a) How many years ago did the image of the
(b) Suppose the star X eaploded and vanished in the
star X we are looking at exist? (Give your answer
year 1380. When we look at the sky at a night
correct to the nearest year.)
in 2003, can we still see the star X?
***********************************************************************************
Level III (Optional)
1. Simplify (1  a p q ) 1  (a q  p  1) 1 , where a, p and q are positive numbers.
(Ans:1)
2.
(a) If xy = 7, then
2 ( x y )
2
2 ( x y )
2
=?
(87AMCSen Ans: 2
28
)
(b) If p  360 , q  5 48 , r  6 36 and s  7 24 , then
arrange p, q, r and s in ascending order.
(89AMCInter Ans: s  r  p  q )
F.3 Mathematics Supplementary Notes
Chapter 2 Laws of Indices
9/2005
P. 10
(c) If 21998  21997  21996  21995  k  21995 , then find the value of k.
(98AHSME Ans:k=3)
3.
Given A  123456789123456786 , B  123456788123456787 . Which product is larger?
(Hint: Let x = 123456786. Then A=(x+3)x, B=?
Ans.B>A)
數學課外閱讀 :
作者 : 李毓佩
出版社: 國際村文庫
內容簡介 :
您想知道“代數”是怎樣被發現的嗎？您想了解“代數”的威力有多大嗎？從
本書中，您可以掌握到“代數”幾千年來，在中外各國的進展過程，同時您會發
覺“代數”在生活中帶給我們有多大的便利，此外“方程博覽會”、“闖關縱橫
談”、“代數萬花筒”也是會讓您愛不釋手的篇章……
資料來源 : 商務網上書店
http://www.cp1897.com.hk/cgi-bin/ncommerce3/ProductDisplay?SECTNBR=18&prmenbr=178&prnbr=9789574521623
F.3 Mathematics Supplementary Notes
Chapter 2 Laws of Indices
9/2005
P. 11
F.3 Mathematics Supplementary Notes
Chapter 2 Laws of Indices
9/2005
P. 12