beams_unit 5 indices

8/9/2019 BEAMS_Unit 5 Indices

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Unit 1:Negative Numbers

UNIT 5

INDICES

B a s i c E s s e n t i a l

A d d i t i o n a l M a t h e m a t i c s S k i l l s

Curriculum Development Division

Ministry of Education Malaysia


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Part C: Indices III 12

1.0 Verifyingmnnm

aa )(13

2.0 Simplifying Numbers Expressed in Index Notation Raised

to a Power 13

3.0 Simplifying Algebraic Terms Expressed in Index Notation Raised

to a Power 14

4.0 Verifying nn

aa

1

15

5.0 Verifying nn aa

1

16

Activity 20

Answers 22


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Basic Essential Additional Mathematics Skills (BEAMS) Module

UNIT 5: Indices

1



PART 1

MODULE OVERVIEW

1. The aim of this module is to reinforce pupils understanding on theconcept of indices.

2. This module aims to provide the basic essential skills for the learning ofAdditional 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 thesub-topic of Indices in Form 4. Teachers can also use this module after

PMR as preparatory work for Form 4 Mathematics and AdditionalMathematics. 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 learningobjectives which can be taught separately. Teachers are advised to use any

sections of the module as and when it is required.


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UNIT 5: Indices

2



LEARNING OBJECTIVES

Upon completion ofPart A, pupils will be able to:

1. express repeated multiplication as an and vice versa;2.

find the value ofa

n

;

3. verify nmnm aaa ;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.

PART A:

INDICES I

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 fromsquares 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.


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UNIT 5: Indices

3



1.0 Expressing Repeated Multiplication Asan

and Vice Versa

(i) 3332

(ii) )4)(4)(4(3)4(

(iii) rrrr 3

(iv) )6)(6()6(2 mmm

2.0 Finding the Value ofan

2 factors of3

3 factors of(4)

3 factors of r

2 factors of(6+m)

32

is read as

three to the power of 2

or

three to the second power.

32

base

81

16

3333

2222

32

32iii)(

125

)5)(5)(5()5()ii(

32

222222)i(

4

44

3

5

LESSON NOTES A

index

(a) What is 24?

(b) What is (1)3?

(c) What is an?


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UNIT 5: Indices

4



3.0 Verifyingnmnm aaa

325

32

213

2

437

43

)1()1(

)]1)(1)(1[()]1)(1[()1()1()iii(

77

)77(777)ii(

22

)2222()222(22)i(

yy

yyyyyyy

4.0 Simplifying Multiplication of Numbers, Expressed In Index Notation with the Same

Base

nmnm aaa

6

515

11

8383

8

14343

3

1

3

1

3

1

3

1)iii(

)5(

)5()5()5()ii(

6

6666)i(


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UNIT 5: Indices

5



3433133

236236

10755255525

15

4

15

4

2

1

5

4

3

2)iii(

30523)ii(

)i(

qpqpqpp

rstrst

nmnmnnmm

5.0 Simplifying Multiplication of Algebraic Terms, Expressed In Index Notation with the

Same Base

6.0 Simplifying Multiplication of Numbers, Expressed In Index Notation with DifferentBases

7.0 Simplifying Multiplication of Algebraic Terms Expressed In Index Notation withDifferent Bases

4133

52323

402011920119

64242

)iv(

)()()()iii(

6632)ii(

)i(

t

s

t

s

t

s

t

s

abababab

wwwww

pppp

4

4

4

4

44

,Conversely

t

s

t

s

t

s

t

s

45423423

17103147331473

312384384

5

3

2

1

5

3

2

1

5

3

2

1

2

1)iii(

75757755)ii(

2323233(i)

Note:

Sum up the indiceswith the samebase.

numbers withdifferent basescannotbesimplified.

555

555

)(

,Conversely

)(

abba

baab


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UNIT 5: Indices

6



1.Find the value of each of the following.(a)

243

3333335

(b) 36

(c) 4)4(

(d)

5

5

1

(e)

3

4

3

(f)

2

5

12

(g) 47 (h)

5

3

2

2.Simplify the following.(a)

5

2323

12

1243

m

mmm

(b) bbb 42 35

(c) 342 3)3(2 xxx

(d) 323 )()2(7 ppp

EXAMPLES & TEST YOURSELF A


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UNIT 5: Indices

7



3.Simplify the following.

(a)576

96434

23

(b)

232

22)3(

(c) 343 )7()7()1(

(d)

232

5

4

3

1

3

1

(e) 423 5522 (f)

7

2

3

2

7

2

3

2223

4.Simplify the following.(a) 2424 1234 gfgf

(b) 232 32)3( srr

(c) 343 )3()7()( vww

(d)

232

5

4

5

1

7

3kkh


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UNIT 5: Indices

8



PART B:

INDICES II

LEARNING OBJECTIVES

Upon completion ofPart B, pupils will be able to:

1. verify nmnm aaa ;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.


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.


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UNIT 5: Indices

9



1.0 Verifying

nmnm

aaa

2. 0 Simplifying Division of Numbers, Expressed In Index Notation with the Same Base

3

5412

54

12

7

310

3

10

4

239239

6

2828

3

333

3(iv)

5

55

5(iii)

7

7777(ii)

4

444(i)

LESSON NOTES B

Note:

1

1

0

0

a

a

aaa

aaaa

m

mmm

mmmm

(a) What is 25 2

5?

(b) What is 20?

(c) What is a0?

nmnm aaa

1

1 1

1

1

/ 1

/ 1

/ 1 1 1

23

23

297

29

352

35

)2()2(

)2)(2(

)2)(2)(2()2()2()iii(

55

55

55555555555)ii(

22

222

2222222)i(

pp

pp

ppppp

1

/ 1

/ 1 1


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UNIT 5: Indices

10



3.0 Simplifying Division of Algebraic Terms, Expressed In Index Notation with the Same

Base

3

8

3

8

3

8(iii)

445

20(ii)

(i)

23

2

3

437

3

7

24646

hhh

h

kkk

k

nnnn

4.0 Simplifying Multiplication of Numbers, Expressed In Index Notation With DifferentBases

5.0 Simplifying Multiplication of Algebraic Terms, Expressed In Index Notation withDifferent Bases

REMEMBER!!!

Numbers with

different basescannot

be simplified.

45

2638

23

68

6

11

6

11

6

415

64

15

6415

5

4

5

4

60

48)ii(

333

3

939(i)

qp

qpqp

qp

k

h

k

h

k

h

kh

hkhh


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UNIT 5: Indices

11



1.Find the value of each of the following.(a)

144

12

121212

2

3535

(b) 999 310

(c)

3

9

8

8

(d)

1218

3

2

3

2

(e)

18

20

)5()5(

(f)

24

1018

3

33


7

512512

q

qqq

(b) 79 84 yy

(c)

8

10

15

35

m

m

(d)

88

1114

2

2

b

b


45

1549

4

59

2

9

2

9

8

36

nm

nmnm

nm

(b)

76

1316

12

64

dc

dc

(c)

34

96

12

64

gf

fgf

(d)

56

489

12

378

vu

uvu

EXAMPLES & TEST YOURSELF B


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UNIT 5: Indices

12



PART C:

INDICES III

LEARNING OBJECTIVES

Upon completion ofPart C of the module, pupils will be able to:

1. derivemnnm aa )(

;

2. simplify(a) numbers;(b) algebraic terms, expressed in index notation raised to a power;

3. verify nn

aa

1 ; and

4. verifynn aa

1

.


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 fromsquares 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.


16/28


UNIT 5: Indices

13



1.0 Verifying

mnnm

aa )(

24

23

8

6

44

33

4

3

4

32

4

3

35391527

555999

595959359

236

33

3323

15

11

15

11

15

11

15

11

15

11

15

11)iii(

2323

23

)23)(23)(23()23()ii(

22

2

22)2()i(

2. 0 Simplifying Numbers Expressed In Index Notation Raised to a Power

245

396

385

3136

3

85

136

(iv)

207

154

2107

534

2)

10(7

53

4(iii)

159

352

539

572

5)

39

7(2(ii)

1210

6210

6)

2(10(i)

mnnmaa )(

LESSON NOTES C


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UNIT 5: Indices

14



3.0 Simplifying Algebraic Terms Expressed In Index Notation Raised to a Power

518

518

273615

23

76535

23

7653

15

20

15

20

15

205

53

5455

3

4

412

412

4

412

4143

44

3

201510

5453525432

105

52552

3

32

3

32

3

32

12

42

12

4)2()v(

32

32

)2(

)2(2)iv(

625

1

625

5

5

1

5

1)iii(

)()ii(

3

3)3((i)

qp

qp

qp

qp

qpp

qp

qpp

n

m

n

m

n

m

n

m

n

m

ba

ba

ba

baba

gfe

gfegfe

x

xx

Note:

A negative number raised toan even power is positive.

A negative number raised toan odd power is negative.


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UNIT 5: Indices

15



4. 0 Verifying nn

aa

1

Alternative Method

n

n

10

110

10

1

100

110

10

1

10

110

110

1010

10010

100010

0001010

2

2

1

1

0

1

2

3

4

n

n

aa

1

352

3

52

2

2

264

2

64

777

1

77777

7777)ii(

3

13

333

1

333333

3333

33)i(

Hint: 100?

1000


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UNIT 5: Indices

16



5.0 Verifying nn aa

1

pp

pp

p

p

pp

p

p

mm

mm

mmm

1

1

1

11

55

1

55 5

1

5

1

5

1

5

1

5

1

55

5

5

1

5

5

1

15

5

15

5

1

21

2

1

2

1

2

2

1

2

2

1

12

2

12

2

1

)iii(

22

222222

22

22

222(ii)

33

333

33

33

333(i)

Take square root on both sides

of the equation.

Note:

mnnm

nn

aa

aa

1

(a) What is 21

4 ?

(b) What is 23

4 ?

(c) What is nm

a ?

nn aa 1


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UNIT 5: Indices

17



1. Find the value of each of the following.(a) (b)

32 ])1[(

(c)

2

2

3

7

2

(d)

32

5

3

(e)

32

5

3

(f)

2. (a) Simplify the following.

(i)

8244246426

32

3232

(ii)

234652

(iii) 5132 44

(iv)

32

5

2

4

3

(v)

23

7

3

4

7

(vi)

4

422

5

43

12

5

327682

22

15

3535

423

2

EXAMPLES & TEST YOURSELF C


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UNIT 5: Indices

18



2. (b) Simplify the following.

(i)

15

155

535153

32

2

))(2(2

x

x

xx

(ii) 674yx

(iii) 3122 ww (iv) 779 84 yy

(v)

2

68

59

9

36

qp

qp

(vi)

3. Simplify the following expressions:

(a)

32

1

2

12

5

5

(b)

1

4

3

(c)

4

23y

x

(d)

51

4

6

2

ts

st

(e)

3

23

12

2 km

nm

(f)

2

63

32

2

8

ba

cab

4423 32 mnnm


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UNIT 5: Indices

19



4. Find the value of each of the following.

(a)

4

6464 331

(b)2

5

100

(c)

43

81 (d)

21

21

273

(e) m

1

m235

110 )()( aaa

(f)

3

4

27

1


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UNIT 5: Indices

20



1. 52

10

44

4

P04 O

34 R 174 T 134

2. 2327 551010 T 514510 O 65510 N 55510 B 614510

3. 222

4

32

D4

22

E 2

2

2

3 N 2

2

4

3 O

3

42

4. xyxy 239 82

M4

27xy A 4

114

x

y L

4

21xy K 2

74

x

y

5. 425 32 A

820 32 N 69 32 T 620 32 S 89 32

6. 4225 nnmm T

87nm U

810nm L67nm E

610nm

Solve the questions to discover the WONDERWORD!

You are given 11 multiple choice questions. Choose the correct answer for each of the question. Use the alphabets for each of the answer to form the WONDERWORD!

ACTIVITY


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UNIT 5: Indices

21



7.

3243

5

2

5

2

5

2

5

2

F12

5

2

A2

5

2

V6

5

2

E5

5

2

8.

5

3

2

4

7

Y

15

10

4

7 R

8

7

4

7 M

8

10

4

7 A

15

7

4

7

9. 36 59525 ba ba

L81515 ba I

835 ba S23

5 ba T 5615 ba

10.

5232

5

2

5

2

3

1

3

1

P105

5

2

3

1

E76

5

2

3

1

I75

5

2

3

1

R106

5

2

3

1

11. 23

76

3

12

qp

qp

Y3

53qp A

534 qp R 99

3

1

qp D

993 qp

Congratulations! You have completed this activity.

1 2 3 4 5 6 7 8 9 10 11

The WONDERWORD IS: ........................................................


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UNIT 5: Indices

22



TEST YOURSELF A:

1.

(a) 243 (b) 216

(c) 256 (d)

3125

1

(e)

64

27

(f)

25

214

(g) 2401 (h)

243

32

2.

(a) 512m (b) 715b

(c) 918x (d) 814p

3.

(a) 576 (b) 288

(c) 823543 (d)

6075

16

(e) 000250 (f)

34983

256

ANSWERS


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UNIT 5: Indices

23



4.

(a) 2412 gf

(b) 2554 sr

(c) 3782764 vw (d) 52125153

144kh

TEST YOURSELF B:

1.

(a) 144 (b) 441531

(c) 144262 (d)

729

64

(e) 25 (f) 81

2.

(a) 7q (b) 22

1y

(c) 2

3

7m

(d) 364b

3.

(a)45

2

9nm

(b) 610

3

16dc

(c) 632 gf

(d) 3714 vu


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UNIT 5: Indices

24



TEST YOURSELF C:

1.(a) (b) 1

(c)

2401

64

(d)

15625

729

5

36

(e)

125

729

5

33

6

(f)

2. (a)

(i) 83224

(ii) 624 52

(iii) (iv)

)5(2

33

2

(v)3

2

4

)3(7

(vi)

2

146

5

)4(3

2. (b)

(i) 1532x (ii)4224

yx

(iii)30

1

w

(iv)

7

14

2

y

(v) 2

16

q

p

(vi) 187162 nm

32768

21677716224

114


28/28


UNIT 5: Indices

3.

(a)

32

1

2

1

5

(b)

3

4

(c)4

8

81x

y

(d)

9

2

3

1

t

s

(e) 3368 nmk (f)

16

64

16

1

b

ca

4.

(a) 4 (b) 000100

(c)

27

1

(d) 9

(e) 5a (f)

81

1

ACTIVITY:

The WONDERWORD is ONEMALAYSIA

beams_unit 5 indices

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