Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae
FURTHER
EXPLORATION
SUGGESTED WEBSITES:
1. http://www.themathpage.com/alg/algebraic-expressions.htm
2. http://www.wtamu.edu/academic/anns/mps/math/mathlab/beg_algebra/beg_alg_tut11_si
mp.htm
3. http://www.helpalgebra.com/onlinebook/simplifyingalgebraicexpressions.htm
4. http://www.tutor.com.my/tutor/daily/eharian_06.asp?h=60104&e=PMR&S=MAT&ft=F
TN
Curriculum Development Division 37
Ministry of Education Malaysia
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae
TEST YOURSELF A: ANSWERS
1. 9a – 5b
3. 6k 2. – 2m – 4n
4. 2p
5. 15 x y 6. 20h 6k
5xy
15
7. 6ab 8. 4(4c d)
7c
3c d
9. x 10. 2
z2
v2
11. 2x 12. 4 2x
5 6x
4 5x
TEST YOURSELF B:
1. – 8n + 3 6. x + y
2. 3q + 1 7. e 2
2 8. n2 m2 mn
3. – 12x2 + 18xy 9. f 2 2 fg
4. – 3b 10. h2 2ih 5i 2
5. p
Curriculum Development Division 38
Ministry of Education Malaysia
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae
TEST YOURSELF C:
1. 3(p 2 – 5) 2. 2(x 2 – 3) 3. x(x – 4)
6. 7m (1 + 2n)
4. m(5m + 12) 5. p(q – 2) 9. 2(x – 3)(x + 3)
12. (3x – 4)(x + 2)
7. (k + 12)(k – 12) 8. (2p – 1)(2p + 1) 15. (2x – 5)(x +1)
18. (2x – 3)(x – 4)
10. (3m + 13)(3m – 13) 11. (2x + 5)(x – 2)
(c) x = 2y – 1
13. (3p + 4)(p – 3) 14. (4p + 1)(p – 1) (f) x = 3y – 4
(c) x 36 y2
16. (2x – 5)(2x – 1) 17. (5p + 6)(p – 1) (f) x y2 1
19. (1 + 3r)(3p + k) 20. (2c – t)(2c – 3w) (c) u fv
TEST YOURSELF D: (b) x 3 y v f
1. (a) x = 2 – y 2 (f) C B
(d) x = 4 – y
2. (a) x = y2 (e) x 5 y BA
(d) x y 12 3
3 39
(b) x 4 y2
3. (a) x 3a (e) x 1 y 2
(d) p 7q 2
2 (b) x y 1
(g) y x y 1
2(x 1)
(e) m p
2n 3
(h) g 4 2l
T2
Curriculum Development Division
Ministry of Education Malaysia
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae
ACTIVITIES
CROSSWORD PUZZLE
RIDDLES
RIDDLE 1
2 31547689
FAN TAST I C
RIDDLE 2
2 13547698
WO N DERFUL
Curriculum Development Division 40
Ministry of Education Malaysia
Basic Essential
Additional Mathematics Skills
UNIT 4
LINEAR EQUATIONS
Unit 1:
Negative Numbers
Curriculum Development Division
Ministry of Education Malaysia
TABLE OF CONTENTS
Module Overview 1
Part A: Linear Equations 2
Part B: Solving Linear Equations in the Forms of x + a = b and x – a = b 6
Part C: Solving Linear Equations in the Forms of ax = b and x = b 9
12
a 15
Part D: Solving Linear Equations in the Form of ax + b = c 18
Part E: Solving Linear Equations in the Form of x + b = c 23
a
Part F: Further Practice on Solving Linear Equations
Answers
Basic Essentials Additional Mathematics (BEAMS) Module
UNIT 4: Linear Equations
MODULE OVERVIEW
1. The aim of this module is to reinforce pupils’ understanding on the concept involved in
solving linear equations.
2. The module is written as a guide for teachers to help pupils master the basic skills
required to solve linear equations.
3. This module consists of six parts and each part deals with a few specific skills.
Teachers may use any parts of the module as and when it is required.
4. Overall lesson notes are given in Part A, to stress on the important facts and concepts
required for this topic.
Curriculum Development Division 1
Ministry of Education Malaysia
Basic Essentials Additional Mathematics (BEAMS) Module
UNIT 4: Linear Equations
PART A:
LINEAR EQUATIONS
LEARNING OBJECTIVES
Upon completion of Part A, pupils will be able to:
1. understand and use the concept of equality;
2. understand and use the concept of linear equations in one unknown; and
3. understand the concept of solutions of linear equations in one unknown
by determining if a numerical value is a solution of a given linear
equation in one unknown.
a. deterTmEinAeCifHaINnuGmAerNicDalLvEalAueRNisIaNsGolSutTioRnAoTfEaGgIivEeSn linear equation
in one unknown;
The concepts of can be confusing and difficult for pupils to grasp. Pupils might
face difficulty when dealing with problems involving linear equations.
Strategy:
Teacher should emphasise the importance of checking the solutions obtained.
Teacher should also ensure that pupils understand the concept of equality and
linear equations by emphasising the properties of equality.
Curriculum Development Division 2
Ministry of Education Malaysia
Basic Essentials Additional Mathematics (BEAMS) Module
UNIT 4: Linear Equations
OVERALL LESSON NOTES
GUIDELINES:
1. The solution to an equation is the value that makes the equation ‘true’. Therefore,
solutions obtained can be checked by substituting them back into the original
equation, and make sure that you get a true statement.
2. Take note of the following properties of equality:
(a) Subtraction Algebra
Arithmetic a=b
8 = (4) (2)
a–c=b–c
8 – 3 = (4) (2) – 3
Algebra
(b) Addition a =; b
Arithmetic
8 = (4) (2) a+c=b+c
8 + 3 = (4) (2) + 3 Algebra
(c) Division a=b
Arithmetic
8=6+2 ab c≠0
8 62 cc
33
(d) Multiplication Algebra
Arithmetic a=b
8 = (6 +2) ac = bc
(8)(3) = (6+2) (3)
Curriculum Development Division 3
Ministry of Education Malaysia
Basic Essentials Additional Mathematics (BEAMS) Module
UNIT 4: Linear Equations
PART A:
LINEAR EQUATIONS
LESSON NOTES
1. An equation shows the equality of two expressions and is joined by an equal sign.
Example: 2 4=7+1
2. An equation can also contain an unknown, which can take the place of a number.
Example: x + 1 = 3, where x is an unknown
A linear equation in one unknown is an equation that consists of only one unknown.
3. To solve an equation is to find the value of the unknown in the linear equation.
4. When solving equations,
(i) always write each step on a new line;
(ii) keep the left hand side (LHS) and the right hand side (RHS) balanced by:
adding the same number or term to both sides of the equation;
subtracting the same number or term from both sides of the equations;
multiplying both sides of the equation by the same number or term;
dividing both sides of the equation by the same number or term; and
(iii) simplify (whenever possible).
5. When pupils have mastered the skills and concepts involved in solving linear equations,
they can solve the questions by using alternative method.
What is solving
an equation?
Solving an equation is like solving a puzzle to find the value of the unknown.
Curriculum Development Division 4
Ministry of Education Malaysia
Basic Essentials Additional Mathematics (BEAMS) Module
UNIT 4: Linear Equations
The puzzle can be visualised by using real life and concrete examples.
1. The equality in an equation can be visualised as the state of equilibrium of a balance.
(a) x + 2 = 5 x=3
2.
x=?
2. The equality in an equation can also be explained by using tiles (preferably coloured tiles).
x xx
xx ++ 22 ==55
x + 2x –+ 2 –=25= –5 2– 2
x =x =33
Curriculum Development Division 5
Ministry of Education Malaysia
Basic Essentials Additional Mathematics (BEAMS) Module
UNIT 4: Linear Equations
PART B:
SOLVING LINEAR EQUATIONS IN
THE FORMS OF
x + a = b AND x – a = b
LEARNING OBJECTIVES
Upon completion of Part B, pupils will be able to understand the concept of
solutions of linear equations in one unknown by solving equations in the
form of:
(i) x + a = b
(ii) x – a = b
where a, b, c are integers and x is an unknown.
TEACHING AND LEARNING STRATEGIES
Some pupils might face difficulty when solving linear equations in one
unknown by solving equations in the form of:
(i) x + a = b
(ii) x – a = b
where a, b, c are integers and x is an unknown.
Strategy:
Teacher should emphasise the idea of balancing the linear equations. When pupils
have mastered the skills and concepts involved in solving linear equations, they
can solve the questions using the alternative method.
Curriculum Development Division 6
Ministry of Education Malaysia
Basic Essentials Additional Mathematics (BEAMS) Module
UNIT 4: Linear Equations
PART B:
SOLVING LINEAR EQUATIONS IN THE FORM OF
x + a = b OR x – a = b
EXAMPLES
Solve the following equations. (ii) x 3 5
(i) x 2 5
Solutions: Subtract 2 from both Alternative Method:
sides of the equation.
(i) x 2 5 x25
x+2–2=5–2 Simplify the LHS. x 52
x=5–2 Simplify the RHS. x3
x=3
(ii) x 3 5 Add 3 to both sides of Alternative Method:
x–3+3=5+3 the equation.
x=5+3 x35
x=8 Simplify the LHS. x 53
Simplify the RHS. x 8
Curriculum Development Division 7
Ministry of Education Malaysia
Basic Essentials Additional Mathematics (BEAMS) Module
UNIT 4: Linear Equations
TEST YOURSELF B
Solve the following equations.
1. x + 1 = 6 2. x – 2 = 4 3. x – 7 = 2
4. 7 + x = 5 5. 5 + x = – 2 6. – 9 + x = – 12
7. –12 + x = 36 8. x – 9 = –54 9. – 28 + x = –78
10. x + 9 = –102 11. –19 + x = 38 12. x – 5 = –92
13. –13 + x = –120 14. –35 + x = 212 15. –82 + x = –197
Curriculum Development Division 8
Ministry of Education Malaysia
Basic Essentials Additional Mathematics (BEAMS) Module
UNIT 4: Linear Equations
PART C:
SOLVING LINEAR EQUATIONS IN
THE FORMS OF
ax = b AND x b
a
LEARNING OBJECTIVES
Upon completion of Part C, pupils will be able to understand the concept of
solutions of linear equations in one unknown by solving equations in the
form of:
(a) ax = b
(b) x b
a
where a, b, c are integers and x is an unknown.
TEACHING AND LEARNING STRATEGIES
Pupils face difficulty when solving linear equations in one unknown by solving
equations in the form of:
(a) ax = b
(b) x b
a
where a, b, c are integers and x is an unknown.
Strategy:
Teacher should emphasise the idea of balancing the linear equations. When pupils
have mastered the skills and concepts involved in solving linear equations, they
can solve the questions using the alternative method.
Curriculum Development Division 9
Ministry of Education Malaysia
Basic Essentials Additional Mathematics (BEAMS) Module
UNIT 4: Linear Equations
PART C:
SOLVING LINEAR EQUATION
ax = b AND x b
a
EXAMPLES
Solve the following equations. (ii) m 4
(i) 3m = 12 3
Solutions:
(i) 3 m = 12 Divide both sides of Alternative Method:
3 m 12 the equation by 3.
33 3m 12
m 12 Simplify the LHS. m 12
3
m=4 Simplify the RHS. 3
m4
(ii) m 4 Multiply both sides of Alternative Method:
3 the equation by 3. m 4
3
m3 43 Simplify the LHS. m 34
3 m 12
Simplify the RHS.
m = 43 10
m = 12
Curriculum Development Division
Ministry of Education Malaysia
Basic Essentials Additional Mathematics (BEAMS) Module
UNIT 4: Linear Equations
TEST YOURSELF C
Solve the following equations.
1. 2p = 6 2. 5k = – 20 3. – 4h = 24
4. 7l 56 5. 8 j 72 6. 5n 60
7. 6v 72 8. 7 y 42 9. 12z 96
10. m 4 11. r = 5 12. w = –7
2 4 8
13. t 8 14. s 9 15. u 6
8 12 5
Curriculum Development Division 11
Ministry of Education Malaysia
Basic Essentials Additional Mathematics (BEAMS) Module
UNIT 4: Linear Equations
PART D:
SOLVING LINEAR EQUATIONS IN
THE FORM OF
ax + b = c
LEARNING OBJECTIVE
Upon completion of Part D, pupils will be able to understand the concept of
solutions of linear equations in one unknown by solving equations in the
form of ax + b = c where a, b, c are integers and x is an unknown.
TEACHING AND LEARNING STRATEGIES
Some pupils might face difficulty when solving linear equations in one
unknown by solving equations in the form of ax + b = c where a, b, c are
integers and x is an unknown.
Strategy:
Teacher should emphasise the idea of balancing the linear equations. When pupils
have mastered the skills and concepts involved in solving linear equations, they
can solve the questions using the alternative method.
Curriculum Development Division 12
Ministry of Education Malaysia
Basic Essentials Additional Mathematics (BEAMS) Module
UNIT 4: Linear Equations
PART D:
SOLVING LINEAR EQUATIONS IN THE FORM OF ax + b = c
EXAMPLES
Solve the equation 2x – 3 = 11.
Solution:
Method 1
2x – 3 = 11 Add 3 to both sides of Alternative Method:
2x – 3 + 3 = 11 + 3 the equation.
2x 3 11
2x = 14 Simplify both sides of 2x 11 3
the equation. 2x 14
2x 14 x 14
22 Divide both sides of 2
the equation by 2. x2
x 14
2 Simplify the LHS.
x=7 Simplify the RHS.
Method 2
2x 3 11 Divide both sides of Alternative Method:
2x 3 11 the equation by 2.
222 2x 3 11
Simplify the LHS. 2x 3 11
x 3 11 222
22 3
Add to both sides x 11 3
x 3 3 11 3 22
22 2 2 2
x 14 of the equation. x 14
2 2
x7 Simplify both sides of
the equation. x7
Curriculum Development Division 13
Ministry of Education Malaysia
Basic Essentials Additional Mathematics (BEAMS) Module
UNIT 4: Linear Equations
TEST YOURSELF D
Solve the following equations.
1. 2m + 3 = 7 2. 3p – 1 = 11 3. 3k + 4 = 10
4. 4m – 3 = 9 5. 4y + 3 = 9 6. 4p + 8 = 11
7. 2 + 3p = 8 8. 4 + 3k = 10 9. 5 + 4x = 1
10. 4 – 3p = 7 11. 10 – 2p = 4 12. 8 – 2m = 6
Curriculum Development Division 14
Ministry of Education Malaysia
Basic Essentials Additional Mathematics (BEAMS) Module
UNIT 4: Linear Equations
PART E
SOLVING LINEAR EQUATIONS IN
THE FORM OF
x bc
a
LEARNING OBJECTIVES
Upon completion of Part E, pupils will be able to understand the concept of
solutions of linear equations in one unknown by solving equations in the form
of x b where a, b, c are integers and x is an unknown.
a
TEACHING AND LEARNING STRATEGIES
Pupils face difficulty when solving linear equations in one unknown by solving
equations in the form of x b where a, b, c are integers and x is an unknown.
a
Strategy:
Teacher should emphasise the idea of balancing the linear equations. When pupils
have mastered the skills and concepts involved in solving linear equations, they
can solve the questions using the alternative method.
Curriculum Development Division 15
Ministry of Education Malaysia
Basic Essentials Additional Mathematics (BEAMS) Module
UNIT 4: Linear Equations
PART E:
SOLVING LINEAR EQUATIONS IN THE FORM OF x b c
a
EXAMPLES
Solve the equation x 4 1 . Add 4 to both sides of Alternative
3 the equation. Method:
Solution: Simplify both sides of x 41
Method 1 the equation. 3
x 4 1 Multiply both sides of x 1 4
3 the equation by 3. 3
x 44 =1+4 x 5
3 Simplify both sides of the 3
equation. x 35
x 5 x 15
3 Multiply both sides of
x 3 53 the equation by 3.
3
x 53 Expand the LHS.
x = 15
Method 2 Simplify both sides of
x 4 3 1 3 the equation.
3
x 3 43 13 Add 12 to both sides of
3 the equation.
x 12 3
x – 12 + 12 = 3 + 12 Simplify both sides of
x 3 12 the equation.
x 15
Curriculum Development Division 16
Ministry of Education Malaysia
Basic Essentials Additional Mathematics (BEAMS) Module
UNIT 4: Linear Equations
TEST YOURSELF E
Solve the following equations.
1. m 3 5 2. b 2 1 3. k 2 7
2 3 3
4. 3 + h = 5 5. 4 + h = 6 6. m 1 2
2 5 4
7. 2 h 5 8. k + 3 = 1 9. 3 h 2
4 6 5
10. 3 – 2m = 7 11. 3 m 7 12. 12 + 5h = 2
2
Curriculum Development Division 17
Ministry of Education Malaysia
Basic Essentials Additional Mathematics (BEAMS) Module
UNIT 4: Linear Equations
PART F:
FURTHER PRACTICE ON SOLVING
LINEAR EQUATIONS
LEARNING OBJECTIVE
Upon completion of Part F, pupils will be able to apply the concept of
solutions of linear equations in one unknown when solving equations of
various forms.
TEACHING AND LEARNING STRATEGIES
Pupils face difficulty when solving linear equations of various forms.
Strategy:
Teacher should emphasise the idea of balancing the linear equations. When pupils
have mastered the skills and concepts involved in solving linear equations, they
can solve the questions using the alternative method.
Curriculum Development Division 18
Ministry of Education Malaysia
Basic Essentials Additional Mathematics (BEAMS) Module
UNIT 4: Linear Equations
PART F:
FURTHER PRACTICE
EXAMPLES
Solve the following equations: Alternative Method:
(i) – 4x – 5 = 2x + 7 4x 5 2x 7
Solution: 4x 2x 7 5
6x 12
Method 1 x 12
4x 5 2x 7 6
x 2
–4x – 2x – 5 = 2x – 2x + 7
6x 5 7 Subtract 2x from both sides of the equation.
6x 5 5 7 5 Simplify both sides of the equation.
6x 12
6x 12 Add 5 to both sides of the equation.
6 6
x 2 Simplify both sides of the equation.
Divide both sides of the equation by –6.
Method 2 Add 5 to both sides of the equation.
4x 5 2x 7 Simplify both sides of the equation.
Subtract 2x from both sides of the equation.
– 4x – 5 + 5 = 2x + 7 + 5 Simplify both sides of the equation.
– 4x = 2x + 12 Divide both sides of the equation by – 6.
– 4x – 2x = 2x – 2x + 12
– 6x = 12
6x 12
6 6
x 2
Curriculum Development Division 19
Ministry of Education Malaysia
Basic Essentials Additional Mathematics (BEAMS) Module
UNIT 4: Linear Equations
(ii) 3(n – 2) – 2(n – 1) = 2 (n + 5) Expand both sides of the equation.
3n – 6 – 2n + 2 = 2n + 10 Simplify the LHS.
n – 4 = 2n + 10
n – 2n – 4 = 2n – 2n + 10 Subtract 2n from both sides of the equation.
– n – 4 = 10
– n – 4 + 4 = 10 + 4 Add 4 to both sides of the equation.
– n = 14
Divide both sides of the equation by – 1.
n 14
1 1
n 14
Alternative Method:
3(n 2) 2(n 1) 2(n 5)
3n 6 2n 2 2n 10
n 4 2n 10
n 14
n 14
Curriculum Development Division 20
Ministry of Education Malaysia
Basic Essentials Additional Mathematics (BEAMS) Module
UNIT 4: Linear Equations
(iii) 2x 3 x 1 3
32
6 2x 3 x 1 6(3) Multiply both sides of the equation by the
3 2 LCM.
6 2x 3 6 x 1 6(3) Expand the brackets.
3 2 Simplify LHS.
2(2x 3) 3(x 1) 18 Add 3 to both sides of the equation.
Divide both sides of the equation by 7.
4x 6 3x 3 18
7x 3 18
7x 3 3 18 3
7x 21
7x 21
77
x3
Alternative Method:
2x 3 x 1 3
32
6 2x 3 x 1 3 6
3 2
2(2x 3) 3(x 1) 18
4x 6 3x 3 18
7x 3 18
7x 18 3
7x 21
x 21
7
x3
Curriculum Development Division 21
Ministry of Education Malaysia
Basic Essentials Additional Mathematics (BEAMS) Module
UNIT 4: Linear Equations
Solve the following equations. TEST YOURSELF F
1. 4x – 5 + 2x = 8x – 3 – x 2. 4(x – 2) – 3(x – 1) = 2 (x + 6)
3. –3(2n – 5) = 2(4n + 7) 4. 3x 9
5. x 2 5 42
23 6 6. x x 2
35
7. y 5 13y 8. x 2 x 1 9
26 3 42
9. 2x 5 3x 4 0 10. 2x 7 4 x 7
68 9 12
Curriculum Development Division 22
Ministry of Education Malaysia
Basic Essentials Additional Mathematics (BEAMS) Module
UNIT 4: Linear Equations
ANSWERS
TEST YOURSELF B: 2. x = 6 3. x = 9
5. x = –7 6. x = –3
1. x = 5 8. x = –45 9. x = –50
4. x = –2 11. x = 57 12. x = –87
7. x = 48 14. x = 247 15. x = –115
10. x = –111
13. x = –107 3. h = –6
6. n = 12
TEST YOURSELF C: 2. k = – 4 9. z = 8
1. p = 3 5. j = – 9 12. w = – 56
4. l = 8 8. y = – 6 15. u = 30
7. v = 12 11. r = 20
10. m = 8 14. s = 108 3. k = 2
13. t = – 64 6. p 3
TEST YOURSELF D: 2. p = 4 4
1. m = 2 5. y 3 9. x = –1
12. m = 1
4. m = 3 2
7. p = 2 8. k = 2 11. k = 15
10. p = −1 11. p = 3 6. m = 12
9. h = 5
TEST YOURSELF E: 10. b = 9 12. h = −2
1. m = 4 5. h = 10
4. h = 4 8. k = −12
7. h = 12 11. m = −8
10. m = −2
Curriculum Development Division 23
Ministry of Education Malaysia
Basic Essentials Additional Mathematics (BEAMS) Module
UNIT 4: Linear Equations
TEST YOURSELF F:
1. x = − 2 2. x = − 17 3. n 1 4. x = 6
14 8. x = 7
5. x = 3 6. x = 15
9. x = −8 10. x = 19 7. y = 3
Curriculum Development Division 24
Ministry of Education Malaysia
Basic Essential
Additional Mathematics Skills
UNIT 5
INDICES
Unit 1:
Negative Numbers
Curriculum Development Division
Ministry of Education Malaysia
TABLE OF CONTENTS
Module Overview 1
Part A: Indices I 2
1.0 Expressing Repeated Multiplication as an and Vice Versa 3
2.0 Finding the Value of an 3
3.0 Verifying am an amn 4
4.0 Simplifying Multiplication of Numbers, Expressed in Index
Notation with the Same Base 4
5.0 Simplifying Multiplication of Algebraic Terms, Expressed in Index
Notation with the Same Base 5
6.0 Simplifying Multiplication of Numbers, Expressed in Index
Notation with Different Bases 5
7.0 Simplifying Multiplication of Algebraic Terms Expressed in Index
Notation with Different Bases 5
Part B: Indices II 8
9
1.0 Verifying a m a n a mn 9
2.0 Simplifying Division of Numbers, Expressed In Index Notation 10
10
with the Same Base 10
3.0 Simplifying Division of Algebraic Terms, Expressed in Index
Notation with the Same Base
4.0 Simplifying Multiplication of Numbers, Expressed in Index
Notation with Different Bases
5.0 Simplifying Multiplication of Algebraic Terms, Expressed in
Index Notation with Different Bases
Part C: Indices III 12
1.0 Verifying (am )n amn 13
13
2.0 Simplifying Numbers Expressed in Index Notation Raised
to a Power
3.0 Simplifying Algebraic Terms Expressed in Index Notation Raised
to a Power 14
4.0 Verifying a n 1
an
15
1 16
5.0 Verifying a n n a
Activity 20
Answers 22
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 5: Indices
MODULE OVERVIEW
1. The aim of this module is to reinforce pupils’ understanding on the
concept of indices.
2. This module aims to provide the basic essential skills for the learning of
PART 1Additional Mathematics topics such as:
Indices and Logarithms
Progressions
Functions
Quadratic Functions
Quadratic Equations
Simultaneous Equations
Differentiation
Linear Law
Integration
Motion Along a Straight Line
3. Teachers can use this module as part of the materials for teaching the
sub-topic of Indices in Form 4. Teachers can also use this module after
PMR as preparatory work for Form 4 Mathematics and Additional
Mathematics. Nevertheless, students can also use this module for self-
assessed learning.
4. This module is divided into three parts. Each part consists of a few learning
objectives which can be taught separately. Teachers are advised to use any
sections of the module as and when it is required.
Curriculum Development Division 1
Ministry of Education Malaysia
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 5: Indices
PART A:
INDICES I
LEARNING OBJECTIVES
Upon completion of Part A, pupils will be able to:
1. express repeated multiplication as an and vice versa;
2. find the value of an;
3. verify am an amn ;
4. simplify multiplication of
(a) numbers;
(b) algebraic terms, expressed in index notation with the same base;
5. simplify multiplication of
(a) numbers; and
(b) algebraic terms, expressed in index notation with different bases.
TEACHING AND LEARNING STRATEGIES
The concept of indices is not easy for some pupils to grasp and hence they
have phobia when dealing with multiplication of indices.
Strategy:
Pupils learn from the pre-requisite of repeated multiplication starting from
squares and cubes of numbers. Through pattern recognition, pupils make
generalisations by using the inductive method.
The multiplication of indices should be introduced by using numbers and
simple fractions first, and then followed by algebraic terms. This is intended
to help pupils build confidence to solve questions involving indices.
Curriculum Development Division 2
Ministry of Education Malaysia
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 5: Indices
LESSON NOTES A
1.0 Expressing Repeated Multiplication As an and Vice Versa
(i) 32 3 3 32 is read as
2 factors of 3 ‘three to the power of 2’
(ii) (4)3 (4)(4)(4) or
‘three to the second power’.
3 factors of (4)
32 index
(iii) r3 r r r
base
3 factors of r
(a) What is 24?
(iv) (6 m)2 (6 m)(6 m) (b) What is (−1)3?
2 factors of (6+m) (c) What is an?
2.0 Finding the Value of an
(i) 25 2 2 2 2 2
32
(ii) ( 5)3 (5)(5)(5)
125
(iii) 2 4 24
3 34
2 2 2 2
3333
16
81
Curriculum Development Division 3
Ministry of Education Malaysia
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 5: Indices
3.0 Verifying am an amn
(i) 23 24 (2 2 2) (2 2 2 2)
27 234
(ii) 7 72 7 (7 7)
73 7 12
(iii ) ( y 1)2 ( y 1)3 [(y 1)(y 1)][(y 1)(y 1)(y 1)]
( y 1)5 ( y 1)23
am an amn
4.0 Simplifying Multiplication of Numbers, Expressed In Index Notation with the Same
Base
(i) 63 64 6 6341
68
(ii) (5)3 (5)8 (5)38
(5)11
(iii) 1 1 5 1 15
3 3 3
1 6
3
Curriculum Development Division 4
Ministry of Education Malaysia
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 5: Indices
5.0 Simplifying Multiplication of Algebraic Terms, Expressed In Index Notation with the
Same Base
(i) p2 p4 p24 p6 (ab)5 a5b5
(ii) 2w9 3w11 w20 6w91120 6w40 Conversely,
a5b5 (ab)5
(iii) (ab)3 (ab)2 ab32 (ab)5 s 4 s4
t t4
(iv) s 3 s s 31 s 4
t t t t Conversely,
s4 s 4
t4 t
6.0 Simplifying Multiplication of Numbers, Expressed In Index Notation with Different
Bases
(i) 34 38 23 348 23 312 23 Note:
(ii) 53 57 714 73 537 7143 510 717 Sum up the indices
(iii) 1 3 1 2 3 4 1 32 3 4 1 5 3 4
with the same
2 2 5 2 5 2 5 base.
numbers with
different bases
cannot be
simplified.
7.0 Simplifying Multiplication of Algebraic Terms Expressed In Index Notation with
Different Bases
(i) m5 m2 n5 n5 m52 n55 m7n10
(ii) 3t 6 2s3 5r 2 30t 6s3r 2
(iii) 2 p 4 p3 1 q3 4 p13q3 4 p4q3
3 5 2 15 15
Curriculum Development Division 5
Ministry of Education Malaysia
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 5: Indices
EXAMPLES & TEST YOURSELF A
1. Find the value of each of the following. (b) 63
(a) 35 3 3 3 3 3
243
(c) (4)4 (d) 1 5
5
(e) 3 3
4 (f) 2 1 2
(g) 74 5
(h) 2 5
3
2. Simplify the following. (b) 5b2 3b4 b
(a) 3m3 4m2 12m32 (d) 7 p3 (2 p2 ) ( p)3
12m5
(c) 2x2 (3x4 ) 3x3
Curriculum Development Division 6
Ministry of Education Malaysia
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 5: Indices
3. Simplify the following. (b) (3)2 23 22
(a) 43 32 64 9
576
(c) (1)3 (7)4 (7)3 (d) 1 2 1 3 4 2
(e) 2 23 52 54 3 3 5
(f) 2 3 2 2 2 2 2
3 7 3 7
4. Simplify the following. (b) (3r)2 2r3 3s2
(a) 4 f 4 3g 2 12 f 4 g 2
(c) (w)3 (7w)4 (3v)3 (d) 3 h2 1 k 3 4 k 2
7 5 5
Curriculum Development Division 7
Ministry of Education Malaysia
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 5: Indices
PART B:
INDICES II
LEARNING OBJECTIVES
Upon completion of Part B, pupils will be able to:
1. verify am an amn ;
2. simplify division of
(a) numbers;
(b) algebraic terms, expressed in index notation with the same base;
3. simplify division of
(a) numbers; and
(b) algebraic terms, expressed in index notation with different bases.
TEACHING AND LEARNING STRATEGIES
Some pupils might have difficulties in when dealing with division of indices.
Strategy:
Pupils should be able to make generalisations by using the inductive method.
The divisions of indices are first introduced by using numbers and simple
fractions, and then followed by algebraic terms. This is intended to help
pupils build confidence to solve questions involving indices.
Curriculum Development Division 8
Ministry of Education Malaysia
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 5: Indices
LESSON NOTES B
1.0 Verifying am an amn (a) What is 25 ÷ 25?
(b) What is 20?
1 11 (c) What is a0?
(i) 25 23 /2 2/ 2/ 2 2 Note:
21 21 21 am am amm a0
22 253
am am am 1
11 am
(ii) 59 52 /5 5/ 5 5 5 5 5 5 5 a0 1
55
11
57 592
11
(iii) (2 p)3 (2 p)2 (2 p)(2 p)(2 p)
1 (2 p)(2 p) 1
(2 p) (2 p)32
am an amn
2. 0 Simplifying Division of Numbers, Expressed In Index Notation with the Same Base
(i) 48 42 482
46
(ii) 79 73 72 7932
74
(iii) 510 5103
53
57
(iv) 312 312 45
34 35
33
Curriculum Development Division 9
Ministry of Education Malaysia
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 5: Indices
3.0 Simplifying Division of Algebraic Terms, Expressed In Index Notation with the Same
Base
(i) n6 n4 n64 n2
(ii) 20k 7 4k 73 4k 4
5k 3
(iii) 8h3 8 h32 8h
3h 2 3 3
4.0 Simplifying Multiplication of Numbers, Expressed In Index Notation With Different
Bases
REMEMBER!!!
Numbers with
different bases cannot
be simplified.
5.0 Simplifying Multiplication of Algebraic Terms, Expressed In Index Notation with
Different Bases
(i) 9h15 3h4k 6 9h15
3h 4k 6
3h154 3h11 h11
k6 3
k6 k6
(ii) 48 p8q 6 4 p83q 62
60 p3q2 5
4 p5q4
5
Curriculum Development Division 10
Ministry of Education Malaysia
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 5: Indices
EXAMPLES & TEST YOURSELF B
1. Find the value of each of the following. (b) 910 93 9
(a) 125 123 1253 (d) 2 18 2 12
3 3
122
144 (f) 318 310
(c) 89 324
83
(e) (5)20
(5)18
2. Simplify the following.
(a) q12 q5 q125 (b) 4 y9 8y7
q7
(c) 35m10 (d) 214b11
15m8 28 b8
3. Simplify the following.
(a) 36m9n5 9 m94n51 (b) 64c16d13
8m4n 2
12c6d 7
9 m5n4
2
(c) 4 f 6 6 fg 9 (d) 8u9 7v8 3u4
12 f 4g3 12u 6v5
Curriculum Development Division 11
Ministry of Education Malaysia
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 5: Indices
PART C:
INDICES III
LEARNING OBJECTIVES
Upon completion of Part C of the module, pupils will be able to:
1. derive (am )n amn ;
2. simplify
(a) numbers;
(b) algebraic terms, expressed in index notation raised to a power;
3. verify an 1 ; and
an
1
4. verify a n n a .
TEACHING AND LEARNING STRATEGIES
The concept of indices is not easy for some pupils to grasp and hence they
have phobia when dealing with algebraic terms.
Strategy:
Pupils learn from the pre-requisite of repeated multiplication starting from
squares and cubes of numbers. Through pattern recognition, pupils make
generalisations by using the inductive method.
In each part of the module, the indices are first introduced using numbers and
simple fractions, and then followed by algebraic terms. This is intended to
help pupils build confidence to solve questions involving indices.
Curriculum Development Division 12
Ministry of Education Malaysia
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 5: Indices
LESSON NOTES C
1.0 Verifying (am )n amn
(i) (23 )2 23 23
233
26 232
(ii) (39 25 )3 (39 25 )(39 25 )(39 25 )
3999 2555
327 215 393 253
(iii ) 113 2 113 113
154 154 154
1133
154 4
116 1132
158 1542
(am )n amn
2. 0 Simplifying Numbers Expressed In Index Notation Raised to a Power
(i) (102)6 102 6 1012
(ii) (27 93)5 27 5 93 5 235 915
(iii) 43 5 (710)2 43 5 710 2 415 720
(iv) 613 3 613 3 639
58 58 3 524
Curriculum Development Division 13
Ministry of Education Malaysia
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 5: Indices
3.0 Simplifying Algebraic Terms Expressed In Index Notation Raised to a Power
(i) (3x2 )5 35 x25
35 x10
(ii) (e2 f 3 g 4 )5 e25 f 35 g 45
e10 f 15g 20
(iii) 1 a3b 4 1 4 a b34 14
5 5
a12b4
54
a12b4
625
1 a12b4
625
(iv) 2m4 5 (2)5 m45
n3 n 35
Note:
(2)5 m20
n15 A negative number raised to
an even power is positive.
32m20
n15 A negative number raised to
an odd power is negative.
m20
32 n15
(v) (2 p3 )5 4 p6q7 25 4 p35 p6 q7
12p3q2 12 p3q2
32 p1563q 72
3
32 p18q5
3
32 p18q5
3
Curriculum Development Division 14
Ministry of Education Malaysia
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 5: Indices
4. 0 Verifying a n 1
an
(i) 34 36 3 3 3 3
333333
1 346 32
32
32 1
32
(ii) 72 75 77
77777
1 725 73
73
a n 1
an
Alternative Method
104 10000 Hint: 1000 100
103 1000 ?
102 100
101 10
100 1
101 1 1
10 101
102 1 1
100 102
10n 1
10n
Curriculum Development Division 15
Ministry of Education Malaysia
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 5: Indices
1
5.0 Verifying a n n a
(i) 3 1 2 1 2 31
2
32
3 1 2 3
2
1 2 Take square root on both sides
of the equation.
32
3
1 1 3
32 32
1
32 3
(ii) 1 5 1 5 21
25 25
1 5 2
25
5 2 1 5 52
5
5 2 1 2 1 2 1 2 1 2 1 52 1
5 5 5 5 5
(a) What is 4 2 ?
1
3
25 5 2
(b) What is 4 2 ?
m
(c) What is a n ?
1 p 1 p
p
(iii ) m mp m1
1 p
p
p m pm
1
mp p m
1 Note:
an n a 1
an n a
m
an n a m
Curriculum Development Division 16
Ministry of Education Malaysia
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 5: Indices
EXAMPLES & TEST YOURSELF C
1. Find the value of each of the following. (b)
[(1)2 ] 3
(a)
25 3 253
215 32768
(c) 23 2 (d) 3 2 3
72
5
(e) 32 3 (f)
5 24
23
2. (a) Simplify the following. (ii) 26 4 53 2
(i) 26 32 4 264 324
224 38
(iii) 42 3 41 5 (iv) 3 2 2 3
4 5
(v) 7 3 3 2
4 7 (vi) 5 2 32 44 4
12 5
Curriculum Development Division 17
Ministry of Education Malaysia
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