Module PMR

Indices 64

CHAPTER 6 : INDICES

NUMBERS IN INDEX FORM

Example: Express in the form of repeated multiplication.

aaaaa =4

1) 5h 2) ( )3h 3)

2

e

d

4) 45

5) ( )31 6)

4

32

7)

2

43

Example: Write in index notation. h x h x h x h x h x h =

8) g x g x g x g 9). ( ) ( ) ( )( ) ( )dd

ddd

10)

n

e

n

e

n

e

11) 7 x 7 x7 x 7 x 7 12) (-11)(-11)(-11) 13)

72

72

72

72

14)

97

97

Example: Evaluate

2222225 =

= 32

15) 310

16) ( )43 17)

2

43

18) 3

41

19)

3

213

20) ( )34.0 21) ( )205.0

Module PMR

Indices 65

LAWS OF INDICES

LAW 1 nmnm aaa +=

Example: Simplify

35 bb

35+= b

8b=

1) 62 gg 2) 693 hhh 3) ( ) ( )32 ee

Example: Simplify

43 22

432 +=

72=

4) 62 55

5) 683 666 6) ( ) ( )45 33

Example: Simplify

32 43 bb ( ) 3243 += b

512b=

7) 62 54 yy 8) 43 310 ss 9) 45832 kk

Example: Simplify

7265 94 shsh ( ) 767594 ++= sh

131236 sh=

10) 232 37 mbm

11) 234 932

vvv 12) papa 443 36

LAW 2 nmnm aaa =

Example: Simplify

47 jj

47= j

3j=

1) 49 nn 2) 25

a

a

3) 344 tt

Example: Simplify 45 42 = 45 - 2 = 43

4) 54 53

5) 24

77

6) 57 1010

Example: Simplify

35 412 aa 35

412

= a

23a=

7) 45 540 gg 8) 28

420

yy

9) 510 2718 nn

Module PMR

Indices 66

Example: Simplify

( )

9

36

36

36

226

12612

z

z

z

zz

=

=

=

+

10) 85 210 uu 11) 24

218

w

w

12) 48923 mm

LAW 3 ( ) nmnm aa =

Example: Simplify

( )15

53

53

hh

h

=

=

1) ( )24k 2) ( ) 52 g 3) ( )27v

Example: Simplify

( )12

42

42

33

3

=

=

4) ( )354 5) ( ) 526 6) ( )357

Example: Simplify ( )

2015

5453

543

22

2

m

m

m

=

=

7) ( )234ed 8) ( ) 42352 9) ( )2643 g

LAW 4 n

n

aa

1=

Example: Simplify

rl

rl1

=

1) kc 2) pm3 3) tdv

Module PMR

Indices 67

Example: Simplify

32

= 321

81

=

4) 34 5) 17 6) 23

Example: Simplify 5

8

414

= 58 44

= ( )584 +

= 584

= 34

= 64

7) 23 818 8) ( )223 1010

1 9) ( )32

8 1m

m

LAW 5 10 =a

Example: 120 =

1) 0

32

=

2) ( )03 = 3) ( )05.4 =

LAW 6 FRACTIONAL INDICES

Example: Rewrite by using the root and power symbol

93

23

9 =

1) 31

64

2) 51

32 3) 32

8

Example: Write in the form of index

53223 5 =

1) 3 8 2) 35 10

3) 16

Module PMR

Indices 68

Example: Evaluate

( )22

4

21

2

21

=

=

1) 31

27 2) 52

32 3) 43

81

Example: Simplify

2

36

32

34

32

34

99

9

99

=

=

=

+

1) 35

32

1010 2) ( ) 4216 gg 3) ( ) 3214 pp

COMBINATION OF LAWS

Example: Simplify

222

2

0

936

936

=

=

=

+

bb

bbb

1) 317 ggg 2) 459

m

mm

3) 435 232 ggg

Simplify ( )

33

3612

31622

3123

442

2

yx

yx

yxyx

yxxy

=

=

=

+

4) ( ) 1322 xyyx 5) ( ) 24243 yxxy 6) ( ) yxxy 322

Evaluate

21

21

81

31

3

31

=

=

7) 21

251

8) 2

1

94

9) 3

1

278

Module PMR

Indices 69

Evaluate

( )( )

321

22222

42

5

49

229

233

=

=

=

=

10) 322 284 11) ( )327 93 12) ( ) 51214 3242

13) 53

41

31

32168

14) 232

327

15) ( ) 223 999

16) ( ) 506 rr

17) ( )2252

44

abba

18) ( ) 2324 666

19) ( ) 523 xx 20) Given .2166 2 =x Find x .

21) Given that .1255 3 =x Find x . 22) If 6413

=y . Find y .

Module PMR

Indices 70

Common Errors

Errors

Correct Steps

1. 22 22 =

( )222 + =

02 = 2

1. 22 22 =

( )222 + =

02 = 1

2. 24 55 =

245 =

25

2. 24 55 =

( )245 =

245 + =

65

3. 452

kkk

= 452 +k

= 3k

3. 452

kkk

= )4(52 +k

= 47+k

= 11k

Questions based on PMR Format

1. Evaluate 31

23

16

2. Evaluate ( )23284 cba

3. Find the value of 32

53

832

Module PMR

Indices 71

4. Given that 21

34

2 8127 =k .Find the value of .k

5. If 34347 = aa , find the value of .a

6. Simplify 65

21

23 pp

7. Simplify 275

333

8. Simplify 3724

aaba

9. Simplify ( ) 521432

prqrqp

10. Simplify 3425

kmmkk

Module PMR

Indices 72

PMR past year questions

2004

1. Given that 162 2 =x , calculate the value of x . ( 2 marks )

2. Simplify ( ) ( ) 9532422 kmkmk . ( 3 marks )

2005

1. Evaluate 32

21

21

8

123 .

( 3 marks )

2. Given ( )( ),333 212 xx = calculate the value of x . ( 2 marks )

2006

1. Simplify 24

kkk

.

( 2 marks )

Module PMR

Indices 73

2. Find the value of 23

21

2 2183 . ( 3 marks )

2007

1. a). Find the value of 21

23

55 . b). Simplify ( ) 234 hhg . ( 3 marks )

2008

1. Simplify 452

m

mm .

( 2 marks )

2. Find the value of a). 13 22 b). ( )3163 32 ( 3 marks )

Module PMR

Indices 74

CHAPTER 6 : INDICES ANSWERS

1. hhhhh 2. ( ) ( ) ( )hhh 3.

e

de

d

4. 5555

5. ( ) ( ) ( )111 6.

32

32

32

32

7.

43

43

8. 4g

9. ( )5d 10.

3

n

e

11. 57 12. ( )311

13. 4

72

14.

2

97

15. 1000 16. 81

17. 169

18. 641

19. 8

343 or

8742 20. 0.064

21. 0.0025

LAW 1 1. 8g 2. 18h 3. ( )5e 4. 85 5. 176 6. ( )93 7. 820y 8. 730s 9. 9

43 k 10.

3421 bm 11. 96v 12. 5718 pa

LAW 2 1. 5n 2. 3a 3. t4 4. 5 5. 72 6. 1210 7. g8 8. 65y

9. 532

n 10. 5u13 11. 69w 12. 12

227

m

LAW 3 1. 8k 2. 10g 3. 14v 4. 154 5. 106 6. 157 7. 68ed 8. 81252 9. 1283 g

LAW 4

1. kc1

2. pm

3 3.

tv

d

4. 341

5. 71

6. 231

7. 8 8. 10 9. 2m

LAW 5 1. 1 2. 1 3. 1

Module PMR

Indices 75

LAW 6 1. 3 64 2. 5 32 3.

23 8

1. 31

8 2. 53

10 3. 21

16 1. 3 2. 4 3. 27

4. 101

5. 7g 6. p

COMBINATION OF LAWS 1. 3g 2. 10m 3. 43g 4. 458 yx 5. 629 yx 6. 31 yx 7.

515 1 = 8.

32

9. 32

10. 128 11. 243 12. 8

13. 32 14. 81 15. 81 16. 5r 17. 834 ba 18. 1 19. x 20. 5=x 21. 6 22. 4

Questions Based on PMR Format 1.

41

2. 3126 cba 3. 32 4. 3

5. 7 6. 3

4

6 p 7. 1031

8. 2b

9. qrp

10. 21

km

PMR QUESTIONS

YEAR SOLUTION AND MARK SCHEME SUB MARK FULL MARK

2004 42 22 =xx or 42 =x 6=x

1 1

2

2004 9568442 kmkkm 9685416 + km

5116 km

1 1 1

3

Module PMR

Indices 76

2005

( )( )

234626

2

6

8

36

2

1

32

3

21

2

32

21

1

1

1

3

2005 212 += xx 122 += xx

3=x

1

1

2

2006 7

)2(14

kk +

1 1

2

2006 ( )( ) 21212122

23

21

2

2233

2293

10842723 23

1

1

1

3

2007 a) 155 or b)

123

3123

ghhorgh

1 1

1

3

2008

morm

mormormorm

12

45263

+

1

1

2

2008 a) 1624 = b) = 3

16313

32

499

21

32 21

=

1

1

1

3