01-indices

Chapter 1: INDICES

1.1 Index Notation

Number in the index notation or in the form of index can be written as:

Base " Index

Example;

a) 4 x 4 = 4$ Index value is 2

Repeated two times Repeated multiplication of the same index value 4

b) 4 x 4 x 4 = 4% Index value is 3
Repeated three times Repeated multiplication of the same index value 4

c) Reminder : " ≠ a x n

Example 1:
Simplify the following repeated multiplication in the form of index ".

a) 0.3 x 0.3 x 0.3 x 0.3
b) (-2) x (-2) x (-2)
c) n x n x n x n x n x n x n x n
Solution:
a) 0.3 x 0.3 x 0.3 x 0.3 = 0.3)

Repeated 4 times

b) (-2) x (-2) x (-2) = (−2)%
Repeated 3 times

Chapter 1: INDICES

c) n x n x n x n x n x n x n x n = /

Repeated 8 times

From the Example 1 solution, a number " means that it has an equal factors of the number a, to
be multiplied together. In general,

" = a x a x a x ….. x a ; ≠ 0
n factor

Example 2 :

Simplify 64 in the form of index with base 2, base 4 and base 8.

Solution:

a) Repeated division

i) Base 2 (64 repeatedly divide by 2 )

n=6 2 64
2 32
Division continues until value 1
2 16
28
24
22

1

Thus, 64 = 21

ii) Base 4 (64 repeatedly divide by 4)

n=3 4 64
4 16
44

1

Thus, 64 = 4%

iii) Base 8 (64 repeatedly divide by 8)
8 64

Chapter 1: INDICES

n=2 88
1

Thus, 64 = 8$
b) Repeated Multiplication

i) Base 2
2x2x2x2x2x2
4
8
16

32
64

Thus, 64 = 21

ii) Base 4
4x4x4
16

64
Thus, 64 = 4%

iii) Base 8
8 x 8 = 64

Thus, 64 = 8$

Chapter 1: INDICES

Example 3:

Simplify %$ in the form of index with base $ .
%3$4 4

Solution:

a) Repeated division

2 32 5 3125

2 16 5 625
5 125
n=5 28 5 25
24
55
22 1

1

Thus, %$ = ( 4$ )4
%3$4

b) Repeated multiplication

$ x $ x $ x $ x $
4 4 4 4 4

4
25

8
125

16
625

32
3125

Thus, %$ = ( $4 )4
%3$4

Chapter 1: INDICES

Example 4 : b) (0.6)%
Find the value of each of the following. 0.6 x 0.6 x 0.6
a) 24 0.36 x 0.6
0.216
2x2x2x2x2
4x2 Thus, (0.6)% = 0.216
8 x2
16 x 2
32

Thus, 24 = 32

Chapter 1: INDICES

1.2 Law of Indices

1. The multiplication of numbers with index notation with the same base can be simplified using
a) Repeated multiplication

b) Addition of indices

2. The division of numbers with index notation and with the same base can be simplified using
a) Repeated multiplication followed by cancellation.

b) Subtraction of indices

3. A number with an index notation which is raised to a power can be simplified using

a) Repeated multiplication

b) Multiplication of indices

4. Negative power of a number is known as negative index.

8" = 3 where n is a positive integer.
9:

5. A fractional index is an index or a power which is a fraction.

; :

: =

<

6. : can be stated as follows

a) ( = ; or ( :; )=

):

b) : = or (: )=

7. Mathematics Tips:
a) = x " = =>"
b) = ÷ " = =8"
c) ? = 1, a ≠ 0
d ) ( =)" = ="

Chapter 1: INDICES

Example 5 : b) Addition of indices
Simplify 2% x 2) using 2% x 2) = 2%>)
a)Repeated multiplication = 2@

2% x 2) = (2 x 2 x 2) x (2 x 2 x 2 x 2)

=2x2x2x2x2x2x2
= 2@

Example 6 :
Given 3A x 34 = 333, what is the value of x ?
Solution:

3A x 34 = 333
3A>4 = 333

Equating the indices:
x + 5 = 11
x = 11 – 5
=6

Example 7 :
Simplify 64 ÷ 6% using

a) Multiplication and cancellation b) Subtraction of indices
64 ÷ 6% = 648%
64 ÷ 6% = 1 A 1 A 1 A 1 A 1 = 6$
1 A 1 A 1

= 6$

Example 8:
Simplify ( 1)% using

a) Repeated multiplication Chapter 1: INDICES
( 1)% = 1 x 1 x 1
= 1>1>1 b) Multiplication of indices
= 3/ ( 1)% = 1A%
= 3/

Example 9:

Simplify each of the following:

a) ( ) ÷ / E

) F

E

b) 625 F

Solution:

a) ( ) ÷ / E = ( ) E ÷ ( / ) FE E = (5) E

) F ) F b) 625 F ) F

= ) A EF ÷ / A FE = 5) A EF

= % x 3 = 5%
"G

= =E =125
"G

Example 10:

Find the value of

a) 5$ x ; x ;

16H 27 E

b) 5$ ÷ 58$

c) JKLG A JME
JNLEMH

Solution:

a) 5$ x ; x ; = 25 x 16 x E 27

16H 27 E

= 25 x 4 x 3

= 300

Chapter 1: INDICES

b) 5$ ÷ 58$ = 5$8(8$)

= 5)

c) JKLG A JME = 4>38/ 18% %8$
JNLEMH

= 8$ % / LEM
JH