Laws of Indices

1
Laws of Indices
2
Mathematics in Workplaces
2.1 Simplifying Algebraic Expressions Involving
Indices
2.2 Zero and Negative Integral Indices
2.3 Simple Exponential Equations
2.4 Different Numeral Systems
2.5 Inter-conversion between Different
Numeral Systems
Chapter Summary
2
Mathematics in Workplaces
Biologist In the 1840s, 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.
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2.1 Simplifying Algebraic Expressions Involving
Indices
A. Law of Index of (am)n
Suppose m and n are positive integers, we have
? amn
If m and n are positive integers, then (am)n ?
amn.
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2.1 Simplifying Algebraic Expressions Involving
Indices
A. Law of Index of (am)n
Example 2.1T
Simplify each of the following expressions. (a) (q
3)x (b) (q3)8 (c) (q2y)5
Solution
(a) (q3)x ? q3 ? x
(b) (q3)8 ? q3 ? 8
(c) (q2y)5 ? q2y ? 5
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2.1 Simplifying Algebraic Expressions Involving
Indices
B. Law of Index of (ab)n
Suppose n is a positive integer, we have
? anbn
If n is a positive integer, then (ab)n ? anbn.
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2.1 Simplifying Algebraic Expressions Involving
Indices
B. Law of Index of (ab)n
Example 2.2T
Simplify each of the following expressions. (a) (1
1u2)2 (b) (3b4)3
Solution
(a) (11u2)2 ? 112u2 ? 2
(b) (3b4)3 ? 33b4 ? 3
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2.1 Simplifying Algebraic Expressions Involving
Indices
8
2.1 Simplifying Algebraic Expressions Involving
Indices
Example 2.3T
Solution
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2.1 Simplifying Algebraic Expressions Involving
Indices
Example 2.4T
Solution
10
2.1 Simplifying Algebraic Expressions Involving
Indices
Example 2.5T
Simplify 64y ? 8x ? 42y.
Solution
64y ? 8x ? 42y
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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
For example, 32 ? 32 ? 32 ? 2 ? 30.
However, if we calculate the actual value of the
expression 32 ? 32, 32 ? 32 ? 9 ? 9 ? 1
We can conclude that 30 ? 1.
Hence, we define the zero index of any non-zero
number as follows
If a ? 0, then a0 ? 1.
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B. Negative Integral Indices
2.2 Zero and 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 ? 5?1.
Hence, we define the negative index of any
non-zero number as follows
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2.2 Zero and Negative Integral Indices
B. Negative Integral Indices
Example 2.6T
Find the values of the following expressions
without using a calculator. (a) 30 ?
25 (b) (?7)?3 ? (?2)0 (c) 5?3 ? (?10)?2
Solution
14
2.2 Zero and Negative Integral Indices
B. Negative Integral Indices
Summarizing the previous results, we have the
following laws of integral indices.
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2.2 Zero and Negative Integral Indices
B. Negative Integral Indices
Example 2.7T
Simplify the following expressions and express
the answers with positive indices. (a) (u2)2(u?1)5
, u ? 0 (b) (3s?1) ? (?s)?4, s ? 0
Solution
16
2.2 Zero and Negative Integral Indices
B. Negative Integral Indices
Example 2.8T
Solution
Alternative Solution
17
2.2 Zero and Negative Integral Indices
B. Negative Integral Indices
Example 2.8T
Solution
18
2.3 Simple Exponential Equations
Consider the equation 2x ? 8.
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 ? 8 2x ? 23 x ? 3
Then simplify the expression using laws of
integral indices if necessary.
For example, (9t)2 ? 81 92t ? 92 2t ?
2 t ? 1
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2.3 Simple Exponential Equations
Example 2.9T
Solution
(a) 103k ? 1000
(b) 2k ? 1
103k ? 103
2k ? 20
3k ? 3
6k ? 6?3
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2.3 Simple Exponential Equations
Example 2.10T
Solution
(b) 2x ? 1 ? 5 ? 2x ? 28
2 ? 2x ? 5 ? 2x ? 28
(2 ? 5) ? 2x ? 28
7 ? 2x ? 28
2x ? 4
2x ? 22
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2.4 Different Numeral Systems
A. Denary System
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.
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2.4 Different Numeral Systems
B. Binary System
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.
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2.4 Different Numeral Systems
C. Hexadecimal System
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.
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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
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2.5 Inter-conversion between Different Numeral
Systems
A. Convert Binary/Hexadecimal Numbers into
Denary Numbers
We can make use of the expanded form to convert
binary/hexadecimal numbers into denary numbers.
It can be done by summing up all the terms in the
expanded form.
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2.5 Inter-conversion between Different Numeral
Systems
A. Convert Binary/Hexadecimal Numbers into
Denary Numbers
Example 2.12T
Convert the following binary numbers into denary
numbers. (a) 111(2) (b) 1001(2)
Solution
27
2.5 Inter-conversion between Different Numeral
Systems
A. Convert Binary/Hexadecimal Numbers into
Denary Numbers
Example 2.13T
Convert the following hexadecimal numbers into
denary numbers. (a) 66(16) (b) 12C(16)
Solution
28
2.5 Inter-conversion between Different Numeral
Systems
B. Convert Denary Numbers into Binary/Hexadecimal
Numbers
We make use of division to convert denary numbers
into binary/hexadecimal numbers.
It can be done by considering all the remainders
in the short division.
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2.5 Inter-conversion between Different Numeral
Systems
B. Convert Denary Numbers into Binary/Hexadecimal
Numbers
Example 2.14T
Convert the denary number 33(10) into a binary
number.
Solution
2 33
16 1
2
8 0
2
2
4 0
2
2 0
1 0
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2.5 Inter-conversion between Different Numeral
Systems
B. Convert Denary Numbers into Binary/Hexadecimal
Numbers
Example 2.15T
Convert the denary number 530(10) into a
hexadecimal number.
Solution
16 530
33 2
16
2 1
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Chapter Summary
2.1 Simplifying Algebraic Expressions Involving
Indices
32
Chapter Summary
2.2 Zero and Negative Integral Indices
33
Chapter Summary
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.
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Chapter Summary
2.4 Different Numeral Systems
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,
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Chapter Summary
2.5 Inter-conversion between Different Numeral
Systems
Inter-conversion of numbers can be done by
division or expressing them in the expanded form.
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2.1 Simplifying Algebraic Expressions Involving
Indices
A. Law of Index of (am)n
Follow-up 2.1
Simplify each of the following expressions. (a) (y
6)2 (b) (yd)2 (c) (y3)2m
Solution
(a) (y6)2 ? y6 ? 2
(b) (yd)2 ? yd ? 2
(c) (y3)2m ? y3 ? 2m
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2.1 Simplifying Algebraic Expressions Involving
Indices
B. Law of Index of (ab)n
Follow-up 2.2
Simplify each of the following expressions. (a) (4
m)3 (b) (13a7)2
Solution
(a) (4m)3 ? 43m3
(b) (13a7)2 ? 132b7 ? 2
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2.1 Simplifying Algebraic Expressions Involving
Indices
Follow-up 2.3
Solution
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2.1 Simplifying Algebraic Expressions Involving
Indices
Follow-up 2.4
Solution
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2.1 Simplifying Algebraic Expressions Involving
Indices
Follow-up 2.5
Simplify each of the following expressions. (a) 25
3x ? 125y ? 54y (b) 162x ? 84x ? 23y
Solution
41
2.2 Zero and Negative Integral Indices
B. Negative Integral Indices
Follow-up 2.6
Find the values of the following expressions
without using a calculator. (a) 100 ?
92 (b) (?4)?1 ? 50 (c) 4?3 ? 6?1
Solution
42
2.2 Zero and Negative Integral Indices
B. Negative Integral Indices
Follow-up 2.7
Simplify the following expressions and express
the answers with positive indices. (a) (h4)?1(h?2)
3, h ? 0 (b) (?k)?5 ? (k4), k ? 0
Solution
43
2.2 Zero and Negative Integral Indices
B. Negative Integral Indices
Follow-up 2.8
Solution
Alternative Solution
44
2.2 Zero and Negative Integral Indices
B. Negative Integral Indices
Follow-up 2.8
Solution
Alternative Solution
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2.3 Simple Exponential Equations
Follow-up 2.9
Solution
(a) 23x ? 64
(b) 82y ? 1
23x ? 26
82y ? 80
3x ? 6
2y ? 0
3y ? 3?4
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2.3 Simple Exponential Equations
Follow-up 2.10
Simplify the following exponential equations. (a)
16y ? 1 ? 26 ? 2y (b) 5x ? 1 ? 2 ? 5x ? 75
Solution
(a) 16y ? 1 ? 26 ? 2y
(b) 5x ? 1 ? 2 ? 5x ? 75
(24)y ? 1 ? 26 ? 2y
5 ? 5x ? 2 ? 5x ? 75
24y ? 4 ? 26 ? 2y
(5 ? 2) ? 5x ? 75
4y ? 4 ? 6 ? 2y
3 ? 5x ? 75
2y ? 2
5x ? 25
5x ? 52
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2.4 Different Numeral Systems
C. Hexadecimal System
Follow-up 2.11
(a) Express 8 ? 102 ? 5 ? 101 ? 3 ? 100 as a
denary number. (b) Express 14 ? 162 ? 0 ? 161 ? 1
? 160 as a hexadecimal number.
Solution
48
2.5 Inter-conversion between Different Numeral
Systems
A. Convert Binary/Hexadecimal Numbers into
Denary Numbers
Follow-up 2.12
Convert the following binary numbers into denary
numbers. (a) 101(2) (b) 10011(2)
Solution
49
2.5 Inter-conversion between Different Numeral
Systems
A. Convert Binary/Hexadecimal Numbers into
Denary Numbers
Follow-up 2.13
Convert the following hexadecimal numbers into
denary numbers. (a) 70(16) (b) 5F3(16)
Solution
50
2.5 Inter-conversion between Different Numeral
Systems
B. Convert Denary Numbers into Binary/Hexadecimal
Numbers
Follow-up 2.14
Convert the following denary numbers into binary
numbers. (a) 26(10) (b) 35(10)
Solution
(a)
(b)
2 26
2 35
13 0
2
17 1
2
6 1
2
8 1
2
2
3 0
4 0
2
1 1
2 0
2
1 0
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2.5 Inter-conversion between Different Numeral
Systems
B. Convert Denary Numbers into Binary/Hexadecimal
Numbers
Follow-up 2.15
Convert the following denary numbers into
hexadecimal numbers. (a) 83(10) (b) 418(10)
Solution
(a)
(b)
16 83
2 418
5 3
26 2
2
1 10