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Transcript of Laws of Indices 2 2.1Simplifying Algebraic Expressions Involving Indices 2.2Zero and Negative...
![Page 1: Laws of Indices 2 2.1Simplifying Algebraic Expressions Involving Indices 2.2Zero and Negative Integral Indices 2.3Simple Exponential Equations Chapter.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e945503460f94b98fd6/html5/thumbnails/1.jpg)
Laws of Indices2
2.1 Simplifying Algebraic Expressions Involving Indices2.2 Zero and Negative Integral Indices
2.3 Simple Exponential Equations
Chapter Summary
Mathematics in Workplaces
2.4 Different Numeral Systems
2.5 Inter-conversion between Different NumeralSystems
![Page 2: Laws of Indices 2 2.1Simplifying Algebraic Expressions Involving Indices 2.2Zero and Negative Integral Indices 2.3Simple Exponential Equations Chapter.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e945503460f94b98fd6/html5/thumbnails/2.jpg)
P. 2
BiologistIn the 1840’s, biologists found that all plants and animals, including humans, are made up of cells.
Cells are created from cell division. Each time a cell division takes place, a parent cell divides into 2 daughter cells. Solving exponential equations like 2n 215 can help biologists determine the growth rate of cells.
Mathematics in Workplaces
![Page 3: Laws of Indices 2 2.1Simplifying Algebraic Expressions Involving Indices 2.2Zero and Negative Integral Indices 2.3Simple Exponential Equations Chapter.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e945503460f94b98fd6/html5/thumbnails/3.jpg)
P. 3
2.1 Simplifying Algebraic Expressions Involving Indices
A. Law of Index of (am)n
Suppose m and n are positive integers, we have
times
)(n
mmmmnm aaaaa
timesn
mmmma
amn
For any positive integers h and k, we have ah ak ah k.
If m and n are positive integers, then (am)n amn.
![Page 4: Laws of Indices 2 2.1Simplifying Algebraic Expressions Involving Indices 2.2Zero and Negative Integral Indices 2.3Simple Exponential Equations Chapter.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e945503460f94b98fd6/html5/thumbnails/4.jpg)
P. 4
Simplify each of the following expressions.(a) (q3)x (b) (q3)8 (c) (q2y)5
2.1 Simplifying Algebraic Expressions Involving Indices
A. Law of Index of (am)n
Example 2.1T
Solution:(a) (q3)x q3 x
q3x
(b) (q3)8 q3 8
q24
(c) (q2y)5 q2y 5
q10y
![Page 5: Laws of Indices 2 2.1Simplifying Algebraic Expressions Involving Indices 2.2Zero and Negative Integral Indices 2.3Simple Exponential Equations Chapter.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e945503460f94b98fd6/html5/thumbnails/5.jpg)
P. 5
)...()...( times times
nn
bbbbaaaa
2.1 Simplifying Algebraic Expressions Involving Indices
B. Law of Index of (ab)n
Suppose n is a positive integer, we have
times
)()()()()(n
n ababababab
anbnGroup the terms of a and b separately.
If n is a positive integer, then (ab)n anbn.
![Page 6: Laws of Indices 2 2.1Simplifying Algebraic Expressions Involving Indices 2.2Zero and Negative Integral Indices 2.3Simple Exponential Equations Chapter.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e945503460f94b98fd6/html5/thumbnails/6.jpg)
P. 6
2.1 Simplifying Algebraic Expressions Involving Indices
B. Law of Index of (ab)n
Example 2.2TSimplify each of the following expressions.(a) (11u2)2 (b) (3b4)3
Solution:(a) (11u2)2 112u2 2
121u4
(b) (3b4)3 33b4
3 27b12
![Page 7: Laws of Indices 2 2.1Simplifying Algebraic Expressions Involving Indices 2.2Zero and Negative Integral Indices 2.3Simple Exponential Equations Chapter.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e945503460f94b98fd6/html5/thumbnails/7.jpg)
P. 7
timesn
n
b
a
b
a
b
a
b
a
b
a
times
times
n
n
bbbb
aaaa
n
n
b
a
2.1 Simplifying Algebraic Expressions Involving Indices
n
ba
C. Law of Index of
is undefined.0
1
If n is a positive integer,
then , where b 0.n
nn
b
a
b
a
When a fraction is multiplied by itself n times,
where b 0 and n is any positive integer, we can simplify the expression as follows:
ba
![Page 8: Laws of Indices 2 2.1Simplifying Algebraic Expressions Involving Indices 2.2Zero and Negative Integral Indices 2.3Simple Exponential Equations Chapter.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e945503460f94b98fd6/html5/thumbnails/8.jpg)
P. 8
2 4
22
4 (b)
d
c
d
c
2.1 Simplifying Algebraic Expressions Involving Indices
n
ba
C. Law of Index of
Example 2.3T
42
n
Simplify each of the following expressions.
(a) , n 0 (b) , d 02
4
d
c
Solution:
8
2
d
c
4
44 22 (a)
nn
4
16
n
![Page 9: Laws of Indices 2 2.1Simplifying Algebraic Expressions Involving Indices 2.2Zero and Negative Integral Indices 2.3Simple Exponential Equations Chapter.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e945503460f94b98fd6/html5/thumbnails/9.jpg)
P. 9
)()2(
2 (a)
32
535
kh
kh2
4
233
7)7( (b)
v
uuv
82
4393
7
7
v
uuv
2.1 Simplifying Algebraic Expressions Involving Indices
n
ba
C. Law of Index of
Example 2.4T
)()2(
232
535
kh
kh2
4
233
7 )7(
v
uuv
Simplify each of the following expressions.
(a) , h and k 0 (b) , v 0
For any positive odd integer m, (1)m 1.For any positive even integer n, (1)n 1.
Solution:
223252 kh 28hk
22
3535
2
2
h
kh
2 42
2 233 33
7)7(
v
uuv
4389237 uvvu77
![Page 10: Laws of Indices 2 2.1Simplifying Algebraic Expressions Involving Indices 2.2Zero and Negative Integral Indices 2.3Simple Exponential Equations Chapter.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e945503460f94b98fd6/html5/thumbnails/10.jpg)
P. 10
2.1 Simplifying Algebraic Expressions Involving Indices
n
ba
C. Law of Index of
Example 2.5TSimplify 64y 8x 42y. Solution:
yxy 2236 )2()2()2( yxy 436 222
yxy 4 3 62 yx 232
64y 8x 42y
Change the numbers to the same base before applying the laws of indices, i.e., write64 26, 8 23 and4 22.
![Page 11: Laws of Indices 2 2.1Simplifying Algebraic Expressions Involving Indices 2.2Zero and Negative Integral Indices 2.3Simple Exponential Equations Chapter.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e945503460f94b98fd6/html5/thumbnails/11.jpg)
P. 11
2.2 Zero and Negative Integral Indices
A. Zero Index
In Book 1A, we learnt that am an am n for m n.
Consider the case when m n: am n a0
However, if we calculate the actual value of the expression 32 32,
32 32 9 9 1
Hence, we define the zero index of any non-zero number as follows:
We can conclude that 30 1.
For example, 32 32 32 2 30.
If a 0, then a0 1.
00 is undefined.
![Page 12: Laws of Indices 2 2.1Simplifying Algebraic Expressions Involving Indices 2.2Zero and Negative Integral Indices 2.3Simple Exponential Equations Chapter.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e945503460f94b98fd6/html5/thumbnails/12.jpg)
P. 12
B. Negative Integral Indices
Consider am an am n. If m n, then m n is negative.
The expression am n has a negative index.For example, 52 53 52 3 51.
However, 52 53 25 125 . 5
1
We can conclude that 51 .5
1
Hence, we define the negative index of any non-zero number as follows:
2.2 Zero and Negative Integral Indices
If a 0 and n is a positive integer,
then .n
n
aa
10n is undefined.
![Page 13: Laws of Indices 2 2.1Simplifying Algebraic Expressions Involving Indices 2.2Zero and Negative Integral Indices 2.3Simple Exponential Equations Chapter.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e945503460f94b98fd6/html5/thumbnails/13.jpg)
P. 13
32123 )a( 50
1)7(
1)2()7( )b(
303
2323
)10(
1
5
1)10(5 )c(
125
100
Example 2.6T
B. Negative Integral Indices
2.2 Zero and Negative Integral Indices
Find the values of the following expressions without using a calculator.(a) 30 25 (b) (7)3 (2)0 (c) 53 (10)2
Solution:
32
1
343
1
1
10
5
1 2
3
5
4
![Page 14: Laws of Indices 2 2.1Simplifying Algebraic Expressions Involving Indices 2.2Zero and Negative Integral Indices 2.3Simple Exponential Equations Chapter.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e945503460f94b98fd6/html5/thumbnails/14.jpg)
P. 14
2.2 Zero and Negative Integral Indices
Summarizing the previous results, we have the following laws of integral indices.
If m and n are integers, then1. am an am n
2. am an am n (where a 0)3. (am)n amn
4. (ab)n anbn
5. (where b 0)
6. a0 1 (where a 0)
7. (where a 0)
nn
aa
1
n
nn
b
a
b
a
B. Negative Integral Indices
![Page 15: Laws of Indices 2 2.1Simplifying Algebraic Expressions Involving Indices 2.2Zero and Negative Integral Indices 2.3Simple Exponential Equations Chapter.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e945503460f94b98fd6/html5/thumbnails/15.jpg)
P. 15
5 12 2
5122
)()( )a(
uu
uu
4
41
)(
13
)()3( )b(
ss
ss
33s
Example 2.7T
2.2 Zero and Negative Integral Indices
B. Negative Integral Indices
Simplify the following expressions and express the answers with positive indices.(a) (u2)2(u1)5, u 0 (b) (3s1) (s)4, s 0
Solution:
Since it is stated that each answer should be written with positive indices, it is incorrect to express the answer as u1.
)5( 4 u1u
u
1
43s
s
1 43 s
![Page 16: Laws of Indices 2 2.1Simplifying Algebraic Expressions Involving Indices 2.2Zero and Negative Integral Indices 2.3Simple Exponential Equations Chapter.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e945503460f94b98fd6/html5/thumbnails/16.jpg)
P. 16
43 23
4
0332
)3(
1)1(2
)3(
)3()2( (a)
yz
y
yz
Example 2.8T
2.2 Zero and Negative Integral Indices
B. Negative Integral Indices
5
23
4
)(
qr
p
Simplify the following expressions and express the answers with positive indices.
(a) , y 0 (b) , p, q and r 04
0332
)3(
)3()(2 y
yz
Solution:
463 )3(2 yz 4463 32 yz
64648 zy
Alternative Solution:
)4()4(3 23
4
0332
)3(1 2
)3(
)3()2(
yz
y
yz
64648 zy
![Page 17: Laws of Indices 2 2.1Simplifying Algebraic Expressions Involving Indices 2.2Zero and Negative Integral Indices 2.3Simple Exponential Equations Chapter.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e945503460f94b98fd6/html5/thumbnails/17.jpg)
P. 17
5
4
235
23
4 )(
)( (b)
p
qr
qr
p
Example 2.8T
2.2 Zero and Negative Integral Indices
B. Negative Integral Indices
5
23
4
)(
qr
p
Simplify the following expressions and express the answers with positive indices.
(a) , y 0 (b) , p, q and r 04
0332
)3(
)3()(2 y
yz
Solution:
1020
15
qp
r
5 4
5 25 3)(
p
qr
20
1015
p
qr
Since it is stated that each answer should be written with positive indices, it is incorrect to express the answer in terms of q 10.
![Page 18: Laws of Indices 2 2.1Simplifying Algebraic Expressions Involving Indices 2.2Zero and Negative Integral Indices 2.3Simple Exponential Equations Chapter.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e945503460f94b98fd6/html5/thumbnails/18.jpg)
P. 18
The variable x of this equation appears as an index.
Such equations are called exponential equations.
Method of solving exponential equations:First express all numbers in index notation with the same base.
For example, 2x 82x 23
x 3
2.3 Simple Exponential Equations
Consider the equation 2x 8.
Then simplify the expression using laws of integral indices if necessary.For example, (9t)2 81
92t 92
2t 2 t 1
(am)n amn
Express both sides as powers of 2
Equate the indices on both sides
![Page 19: Laws of Indices 2 2.1Simplifying Algebraic Expressions Involving Indices 2.2Zero and Negative Integral Indices 2.3Simple Exponential Equations Chapter.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e945503460f94b98fd6/html5/thumbnails/19.jpg)
P. 19
2.3 Simple Exponential Equations
Example 2.9T
216
1Simplify the following exponential equations.(a) 103k 1000 (b) 2k 1 (c) 6k
Solution:(a) 103k 1000103k 103
3k 3 k 1
(b) 2k 12k 20
k 0
k 3
(c) 6k
216
1
6k
36
1
6k 63
![Page 20: Laws of Indices 2 2.1Simplifying Algebraic Expressions Involving Indices 2.2Zero and Negative Integral Indices 2.3Simple Exponential Equations Chapter.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e945503460f94b98fd6/html5/thumbnails/20.jpg)
P. 20
1 2 6
1 6
)7(
17
49
17 )a(
mm
mm
2.3 Simple Exponential Equations
Example 2.10TSimplify the following exponential equations.
(a) (b) 2x 1 5 2x 281
6
49
17
m
m
Solution:
8226
)22(6
mmmmm
2 2 6
7
17
m
m
)2 2( 6 77 mm
(b) 2x 1 5 2x 28 2 2x 5 2x 28 (2 5) 2x 28 7 2x 28 2x 4 2x 22
x 2
Express all numbers inindex notation with thesame base.
Apply the techniques ofsolving linear equationswith one unknown.
![Page 21: Laws of Indices 2 2.1Simplifying Algebraic Expressions Involving Indices 2.2Zero and Negative Integral Indices 2.3Simple Exponential Equations Chapter.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e945503460f94b98fd6/html5/thumbnails/21.jpg)
P. 21
The most commonly used numeral system today is the denary system.
Numbers in this system are called denary numbers.
The denary system consists of 10 basic numerals: ‘0’, ‘1’, ‘2’, ‘3’, ‘4’, ‘5’, ‘6’, ‘7’, ‘8’ and ‘9’.
Consider the expanded form of 236 with base 10:
236 = 2 102 + 3 101 + 6 100
The numbers 102, 101 and 100 are the place values of the corresponding positions/digits of a number.
The place values of numbers in this system differ by powers of 10.
A. Denary System
2.4 Different Numeral Systems
![Page 22: Laws of Indices 2 2.1Simplifying Algebraic Expressions Involving Indices 2.2Zero and Negative Integral Indices 2.3Simple Exponential Equations Chapter.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e945503460f94b98fd6/html5/thumbnails/22.jpg)
P. 22
Another commonly used numeral system is the binary system.
Numbers in this system are called binary numbers.
The binary system consists of only 2 numerals: ‘0’ and ‘1’.
For example, the expanded form of 1011(2) is:
1011(2) = 1 23 + 0 22 + 1 21 + 1 20
The numbers 23, 22, 21 and 20 are the place values of the corresponding positions/digits of a number.
The place values of the digits in a binary number differ by powers of 2.
2.4 Different Numeral Systems
B. Binary System
![Page 23: Laws of Indices 2 2.1Simplifying Algebraic Expressions Involving Indices 2.2Zero and Negative Integral Indices 2.3Simple Exponential Equations Chapter.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e945503460f94b98fd6/html5/thumbnails/23.jpg)
P. 23
Another commonly used numeral system is the hexadecimal system.
Numbers in this system are called hexadecimal numbers.
The hexadecimal system consists of 16 numerals and letters: ‘0’, ‘1’, ‘2’, ‘3’, ‘4’, ‘5’, ‘6’, ‘7’, ‘8’, ‘9’, ‘A’, ‘B’, ‘C’, ‘D’, ‘E’ and ‘F’.The letters A to F represent the values 10(10) to 15(10) respectively.
For example, the expanded form of 13A(16) is:
13A(16) = 1 162 + 3 161 + 10 160
The numbers 162, 161 and 160 are the place values of the corresponding positions/digits of a number.
The place values of the digits in a hexadecimal number differ by powers of 16.
2.4 Different Numeral Systems
C. Hexadecimal System
![Page 24: Laws of Indices 2 2.1Simplifying Algebraic Expressions Involving Indices 2.2Zero and Negative Integral Indices 2.3Simple Exponential Equations Chapter.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e945503460f94b98fd6/html5/thumbnails/24.jpg)
P. 24
)2(012 101212021 )a(
)10(012 490100109104 )b(
2.4 Different Numeral Systems
C. Hexadecimal System
Example 2.11T(a) Express 1 22 0 21 1 20 as a binary number.(b) Express 4 102 9 101 0 100 as a denary number.Solution:
![Page 25: Laws of Indices 2 2.1Simplifying Algebraic Expressions Involving Indices 2.2Zero and Negative Integral Indices 2.3Simple Exponential Equations Chapter.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e945503460f94b98fd6/html5/thumbnails/25.jpg)
P. 25
It can be done by summing up all the terms in the expanded form.
A. Convert Binary/Hexadecimal Numbers into Denary Numbers
2.5 Inter-conversion between Different Numeral Systems
We can make use of the expanded form to convert binary/hexadecimal numbers into denary numbers.
![Page 26: Laws of Indices 2 2.1Simplifying Algebraic Expressions Involving Indices 2.2Zero and Negative Integral Indices 2.3Simple Exponential Equations Chapter.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e945503460f94b98fd6/html5/thumbnails/26.jpg)
P. 26
012)2( 212121111 (a)
0123)2( 212020211001 (b)
2.5 Inter-conversion between Different Numeral Systems
A. Convert Binary/Hexadecimal Numbers into Denary Numbers
Example 2.12TConvert the following binary numbers into denary numbers.(a) 111(2)
(b) 1001(2)
Solution:
124
18
)10(7
)10(9
![Page 27: Laws of Indices 2 2.1Simplifying Algebraic Expressions Involving Indices 2.2Zero and Negative Integral Indices 2.3Simple Exponential Equations Chapter.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e945503460f94b98fd6/html5/thumbnails/27.jpg)
P. 27
01)16( 16616666 )a(
012)16( 1612162161C12 )(b
2.5 Inter-conversion between Different Numeral Systems
A. Convert Binary/Hexadecimal Numbers into Denary Numbers
Example 2.13TConvert the following hexadecimal numbers into denary numbers.(a) 66(16)
(b) 12C(16)
Solution:
696 102
1232256 300
![Page 28: Laws of Indices 2 2.1Simplifying Algebraic Expressions Involving Indices 2.2Zero and Negative Integral Indices 2.3Simple Exponential Equations Chapter.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e945503460f94b98fd6/html5/thumbnails/28.jpg)
P. 28
2.5 Inter-conversion between Different Numeral Systems
B. Convert Denary Numbers into Binary/Hexadecimal Numbers
It can be done by considering all the remainders in the short division.
We make use of division to convert denary numbers into binary/hexadecimal numbers.
![Page 29: Laws of Indices 2 2.1Simplifying Algebraic Expressions Involving Indices 2.2Zero and Negative Integral Indices 2.3Simple Exponential Equations Chapter.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e945503460f94b98fd6/html5/thumbnails/29.jpg)
P. 29
2.5 Inter-conversion between Different Numeral Systems
B. Convert Denary Numbers into Binary/Hexadecimal Numbers
Example 2.14TConvert the denary number 33(10) into a binary number.
Solution:2 33
16 … 1
8 … 0
2
2
2 4 … 0
2 2 … 0
1 … 0
33(10) 100001
(2)
![Page 30: Laws of Indices 2 2.1Simplifying Algebraic Expressions Involving Indices 2.2Zero and Negative Integral Indices 2.3Simple Exponential Equations Chapter.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e945503460f94b98fd6/html5/thumbnails/30.jpg)
P. 30
2.5 Inter-conversion between Different Numeral Systems
B. Convert Denary Numbers into Binary/Hexadecimal Numbers
Example 2.15TConvert the denary number 530(10) into a hexadecimal number.
Solution:16 530
33 … 216
2 … 1
530(10) 212(16)
![Page 31: Laws of Indices 2 2.1Simplifying Algebraic Expressions Involving Indices 2.2Zero and Negative Integral Indices 2.3Simple Exponential Equations Chapter.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e945503460f94b98fd6/html5/thumbnails/31.jpg)
P. 31
Chapter Summary
2.1 Simplifying Algebraic Expressions Involving Indices
n
nn
b
a
b
a
For positive integers m and n,
1. (am)n amn.
2. (ab)n anbn.
3. , where b 0.
![Page 32: Laws of Indices 2 2.1Simplifying Algebraic Expressions Involving Indices 2.2Zero and Negative Integral Indices 2.3Simple Exponential Equations Chapter.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e945503460f94b98fd6/html5/thumbnails/32.jpg)
P. 32
Chapter Summary
2.2 Zero and Negative Integral Indices
na
1
For any non-zero number a and positive integer n,
1. a0 1.
2. an .
![Page 33: Laws of Indices 2 2.1Simplifying Algebraic Expressions Involving Indices 2.2Zero and Negative Integral Indices 2.3Simple Exponential Equations Chapter.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e945503460f94b98fd6/html5/thumbnails/33.jpg)
P. 33
2.3 Simple Exponential Equations
When solving exponential equations, first express all numbers in index notation with the same base, then simplify using the laws of integral indices.
Chapter Summary
![Page 34: Laws of Indices 2 2.1Simplifying Algebraic Expressions Involving Indices 2.2Zero and Negative Integral Indices 2.3Simple Exponential Equations Chapter.](https://reader038.fdocuments.net/reader038/viewer/2022110101/56649e945503460f94b98fd6/html5/thumbnails/34.jpg)
P. 34
System Binary Denary Hexadecimal
Digits used
0, 1 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F
Place values
20, 21, … 100, 101, … 160, 161, …
2.4 Different Numeral Systems
Chapter Summary
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P. 35
Inter-conversion of numbers can be done by division or expressing them in the expanded form.
2.5 Inter-conversion between Different Numeral Systems
Chapter Summary