This document provides examples and explanations of laws of indices. It includes expressing numbers in index form, writing numbers in index notation, evaluating expressions using laws of indices, and simplifying combinations of indices. Examples range from single term expressions to more complex expressions combining multiple laws of indices. The document aims to teach readers how to manipulate expressions involving indices and apply the laws of indices.

1. Module PMR
CHAPTER 6 : INDICES
NUMBERS IN INDEX FORM
Example:
Express in the form
of repeated
multiplication.
aaaaa ×××=4
1) 5
h 2) ( )3
h−
3)
2






e
d
4) 4
5 5) ( )3
1−
6)
4
3
2






7)
2
4
3






−
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) 























7
2
7
2
7
2
7
2
14) 





−





−
9
7
9
7
Example:
Evaluate
2222225
××××=
= 32
15) 3
10 16) ( )4
3−
17)
2
4
3






18)
3
4
1





−
19)
3
2
1
3 




 20) ( )3
4.0 21) ( )2
05.0−
LAWS OF INDICES
Indices 64

2. Module PMR
LAW 1 nmnm
aaa +
=×
Example:
Simplify
35
bb ×
35+
=b
8
b=
1) 62
gg × 2) 693
hhh ×× 3) ( ) ( )32
ee −×−
Example:
Simplify
43
22 ×
43
2 +
=
7
2=
4) 62
55 × 5) 683
666 ×× 6)( ) ( )45
33 −×−
Example:
Simplify
32
43 bb ×
( ) 32
43 +
×= b
5
12b=
7) 62
54 yy × 8) 43
310 ss ×
9)
45
8
3
2 kk ×
Example:
Simplify
7265
94 shsh ××
( ) 7675
94 ++
×= sh
1312
36 sh=
10) 232
37 mbm ××
11)
234
9
3
2
vvv ×× 12) papa ××− 443
36
LAW 2 nmnm
aaa −
=÷
Example:
Simplify
47
jj ÷
47−
= j
3
j=
1) 49
nn ÷
2) 2
5
a
a 3) 34
4 tt ÷
Example:
Simplify 45
÷ 42
= 45 - 2
= 43
4) 54
÷ 53
5) 2
4
7
7 6) 57
1010 −
÷
Example:
Simplify
35
412 aa ÷
35
4
12 −
= a
2
3a=
7) 45
540 gg ÷
8) 2
8
4
20
y
y 9) 510
2718 nn ÷
Example:
Simplify
10) 85
210 −
÷ uu
11) 2
4
2
18
−
w
w 12)
48
9
2
3 −
÷ mm
Indices 65

3. Module PMR
( )
9
36
36
36
2
2
6
12
612
z
z
z
zz
=
=
=
÷
+
−−
−
LAW 3 ( ) nmnm
aa ×
=
Example:
Simplify
( )
15
53
53
h
h
h
=
= ×
1) ( )24
k 2) ( ) 52 −
g 3) ( )27−
v
Example:
Simplify
( )
12
42
42
3
3
3
=
= ×
4) ( )35
4 5) ( ) 52
6
−
6) ( )35
7−
Example:
Simplify
( )
2015
5453
543
2
2
2
m
m
m
=
= ××
7) ( )234
ed 8) ( ) 423
52
−
9) ( )264
3 −
g
LAW 4 n
n
a
a
1
=−
Example:
Simplify
r
l −
r
l
1
=
1) k
c−
2) p
m−
3 3) t
dv−
Example:
Simplify
3
2−
= 3
2
1
4) 3
4−
5) 1
7−
6) 2
3−
Indices 66

4. Module PMR
8
1
=
Example:
Simplify
5
8
4
1
4 ×
= 58
44 −
×
= ( )58
4 −+
= 58
4 −
= 3
4
= 64
7) 2
3
8
1
8 × 8) ( )22
3
10
10
1
× 9)
( )32
8 1
m
m ×
LAW 5 10
=a
Example:
120
=
1)
0
3
2






=
2) ( )0
3− = 3) ( )0
5.4 =
LAW 6 FRACTIONAL INDICES
Example:
Rewrite by using
the root and power
symbol
9
3
2
3
9 =
1) 3
1
64 2) 5
1
32 3) 3
2
8
Example:
Write in the form of
index
53
22
3
5 =
1) 3
8 2)
3
5
10 3) 16
Example:
Evaluate
1) 3
1
27 2) 5
2
32 3) 4
3
81
Indices 67

5. Module PMR
( )
2
2
4
2
1
2
2
1
=
=
Example:
Simplify
2
3
6
3
2
3
4
3
2
3
4
9
9
9
99
=
=
=
×
+
1) 3
5
3
2
1010 ÷ 2) ( ) 42
1
6
gg × 3) ( ) 32
1
4 −−
÷ pp
COMBINATION OF LAWS
Example:
Simplify
2
2
2
2
0
936
936
=
=
=
÷×
−+
b
b
bbb
1) 317 −
×÷ ggg
2) 4
59
m
mm × 3) 435
232 ggg ÷×
Simplify
( )
33
3612
31622
3123
4
4
2
2
yx
yx
yxyx
yxxy
=
=
÷=
÷
−+
−
−
4) ( ) 132
2 −
÷xyyx 5) ( ) 2424
3 −−
× yxxy 6) ( ) yxxy 322
÷
Evaluate
2
1
2
1
8
1
3
1
3
3
1
=














=






7)
2
1
25
1





 8)
2
1
9
4





 9)
3
1
27
8






Evaluate 10) 322
284 ÷× 11) ( )327
93 ×−
12) ( ) 5
1
2
1
4
3242 ÷×
Indices 68

6. Module PMR
( )
( )
32
1
2
22
22
42
5
49
229
233
=
=
×=
×=
×
−
−
−
−
13) 5
3
4
1
3
1
32168 ×× 14) 23
2
327 −
÷
15) ( ) 223
999 −
×÷ 16) ( ) 506
rr ÷
17)
( )
2
252
4
4
ab
ba 18) ( ) 2324
666 ÷×−
19) ( ) 523
xx ÷
−−
20) Given .2166
2
=
−x
Find x .
21) Given that .1255 3
=−x
Find x .
22) If
64
13
=−
y . Find y .
Common Errors
Indices 69

7. Module PMR
Errors Correct Steps
1. 22
22 −
×
= ( )22
2 −+
= 0
2
= 2
1. 22
22 −
×
= ( )22
2 −+
= 0
2
= 1
2. 24
55 −
÷
= 24
5 −
= 2
5
2. 24
55 −
÷
= ( )24
5 −−
= 24
5 +
= 6
5
3. 4
52
−
×
k
kk
= 452 −+
k
= 3
k
3. 4
52
−
×
k
kk
= )4(52 −−+
k
= 47+
k
= 11
k
Questions based on PMR Format
1. Evaluate
3
1
2
3
16
−








2. Evaluate ( )2
3
284 −
cba
3. Find the value of 3
2
5
3
832
−
÷
4. Given that 2
1
3
4
2
8127 ÷=k .Find the value of .k
Indices 70

8. Module PMR
5. If 34347
=× −
aa , find the value of .a
6. Simplify 6
5
2
1
23 pp ×
7. Simplify 27
5
33
3
−
−
×
8. Simplify 372
4
−
×× aab
a
9. Simplify
( ) 521
432
prq
rqp
×
××
−
−
10. Simplify 34
25
−
−
×
××
km
mkk
PMR past year questions
2004
1. Given that 162 2
=−x
, calculate the value of x .
( 2 marks )
Indices 71

9. Module PMR
2. Simplify ( ) ( ) 953242
2 kmkmk ÷× .
( 3 marks )
2005
1. Evaluate
3
2
2
1
2
1
8
123 ×
.
( 3 marks )
2. Given ( )( ),333 212 xx
=−
calculate the value of x .
( 2 marks )
2006
1. Simplify 2
4
−
×
k
kk
.
( 2 marks )
2. Find the value of 2
3
2
1
2
2183 ×× .
( 3 marks )
Indices 72

10. Module PMR
2007
1. a). Find the value of 2
1
2
3
55 ÷ .
b). Simplify ( ) 234
hhg × .
( 3 marks )
2008
1. Simplify 4
52
m
mm ×−
.
( 2 marks )
2. Find the value of
a). 13
22 −
÷
b). ( )3
1
63
32 ×−
( 3 marks )
CHAPTER 6 : INDICES
ANSWERS
1. hhhhh ×××× 2. ( ) ( ) ( )hhh −×−×−
3. 





×





e
d
e
d 4. 5555 ×××
Indices 73

11. Module PMR
5. ( ) ( ) ( )111 −×−×−
6. 





×





×





×





3
2
3
2
3
2
3
2
7. 





−×





−
4
3
4
3 8. 4
g
9. ( )5
d−
10.
3






n
e 11. 5
7 12. ( )3
11−
13.
4
7
2






14.
2
9
7






−
15. 1000 16. 81
17.
16
9
18.
64
1
− 19.
8
343
or
8
7
42
20. 0.064
21. 0.0025
LAW 1
1. 8
g 2. 18
h 3. ( )5
e− 4. 8
5
5. 17
6 6. ( )9
3− 7. 8
20y 8. 7
30s
9.
9
4
3
k 10. 34
21 bm 11. ( )35
7−
12. 57
18 pa−
LAW 2
1. 5
n 2. 3
a 3. t4 4. 5
5. 72
6. 12
10 7. g8 8. 6
5y
9.
5
3
2
n
10. 5 u13
11. 6
9w
12.
12
2
27
m
LAW 3
1. 8
k 2. 10−
g 3. 14−
v
4. 15
4 5. 10
6−
6. 15
7−
7. 68
ed 8. 812
52 −−
9. 128
3 −
g
LAW 4
1. k
c
1
2. p
m
3
3. t
v
d
4. 3
4
1
5.
7
1
6. 2
3
1
7. 8 8. 10 9. 2
m
LAW 5
1. 1 2. 1 3. 1
LAW 6
1. 3
64 2. 5
32 3.
2
3
8
1. 3
1
8 2. 5
3
10 3. 2
1
16
1. 3 2. 4 3. 27
4.
10
1 5. 7
g 6. p
COMBINATION OF LAWS
Indices 74

12. Module PMR
1. 3
g 2. 10
m 3. 4
3g 4. 45
8 yx
5. 62
9 yx−
6. 31
yx−
7.
5
1
5 1
=−
8.
3
2
9.
3
2 10. 128 11. 243 12. 8
13. 32 14. 81 15. 81 16. 5−
r
17. 83
4 ba 18. 1 19. x 20. 5=x
21. 6 22. 4
Questions Based on PMR Format
1.
4
1 2. 3126 −
cba 3. 32 4. 3
5. 7 6. 3
4
6 p 7. 10
3
1 8. 2−
b
9. qr
p
10. 2
1
km
PMR QUESTIONS
YEAR SOLUTION AND MARK SCHEME
SUB
MARK
FULL
MARK
2004 42
22 =−x
x or 42 =−x
6=x
1
1 2
2004 956844
2 kmkkm ÷×
96854
16 −+−
km
51
16 km −
1
1
1 3
2005
( )
( )
2
3
4
6
2
6
2
6
8
36
2
1
3
2
3
2
1
2
3
2
2
1










1
1
1
3
2005 212 +=− xx
122 +=− xx
3=x
1
1 2
Indices 75

13. Module PMR
2006
7
)2(14
k
k −−+
1
1 2
2006
( )
( ) 2
1
2
1
2
1
22
2
3
2
1
2
2233
2293
×××
×××
108
427
23 23
×
×
1
1
1
3
2007 a) 1
55 or
b) 123
3123
gh
horgh
1
1
1
3
2008
m
orm
mormormorm
12
45263
−
−+−−
1
1
2
2008 a) 1624
=
b) =
××−
× 3
1
6
3
1
3
32
4
9
9
2
1
32 21
=×
×−
1
1
1
3
Indices 76