The document explains the concepts of indices, laws of indices, and their applications, including zero and negative indices. It also covers standard form, fractional indices, and provides several worked examples for clarification. Exponential equations and how to manipulate numbers in standard form are emphasized throughout.

1. INDICES AND
STANDARD FORM
Mathematics

2. INDICES
2
> The index refers to the power to which a number is raised.
> In the example 5³, the number 5 is raised to the power of 3, which
means 5 x 5 x 5.
> The 3 is known as the index and the 5 is known as the base.
> Indices is the plural form of index.
Here are some examples:
> 33 = 3 x 3 x 3 = 27
> 74 = 7 x 7 x 7 x 7 = 2401
> 31 = 3

3. LAWS OF INDICES
3
When working with numbers involving indices there are
three basic laws which can be applied. These are:
1. am x an = am+n
2. am ÷ an = am-n
3. (am)n = amxn

4. Worked examples
a. Simplify 43 x 44.
43 x 44 = 4(3+4) = 47
b. Evaluate (42)3.
(42)3 = 42x3 = 46 = 4096
c. Simplify 2 x 2 x 2 x 5 x 5 using indices.
2 x 2 x 2 x 5 x 5 = 23 x 52
a. Write out 43 x 63 in full.
43 x 63 = 4 x 4 x 4 x 6 x 6 x 6
4

5. The Zero Index
5
> The zero index indicates
that a number is raised
to the power 0.
> A number that is raised
to the power 0 is equal to
1.
am ÷ an = am-n therefore
= a0
However,
therefore a0 = 1
m
m
m
m
a
a
a 

1

m
m
a
a

6. Negative Indices
6
> A negative index
indicates that a number
is being raised to a
negative power: e.g 4-3.
> Another law of indices
states that
This can be proved as follows:
a-m = a0-m
=
=
therefore
m
a
a0
m
m
a
a
1


m
m
a
a
1


m
a
1

7. Exponential equations
> Equations that involve indices as unknown are known as
exponential equations.
> Worked examples:
a. Find the value of x if 2x = 32
32 can be expressed as a power of 2, 32 = 25
Therefore 2x = 25 -> x = 5
b. Find the value of m if 3(m-1) = 81
81 can be expressed as a power of 3, 81 = 34
Therefore 3(m-1) = 34 m – 1 = 4
m = 5
7

8. STANDARD FORM
> Standard form is also known as standard index form or
sometimes as scientific notation.
> Positive indices and large numbers
- 100 = 1 x 102
- 1000 = 1 x 103
- 10000 = 1 x 104
- 3000 = 3 x 103
- For a number to be in standard form it must take the form
A x 10n where the index n is a positive or negative integer
and A must lie in the range 1 ≤ A < 10.
8

9. Worked examples
a. Write 72000 in standard form.
72000 = 7.2 x 104
b. Write 4 x 104 as an ordinary number.
4 x 104 = 4 x 10000 = 40000
c. Multiply the following and write your answer in standard form.
600 x 4000 = 2400000 = 2.4 x 106
d. Multiply the following and write your answer in standard form.
(2.4 x 104) x ( 5 x 107) = 12 x 1011
= 1.2 x 1012 (standard form)
9

10. Worked examples
e. Divide the following and write your answer in standard form:
(6.4 x 107) ÷ (1.6 x 103) = 4 x 104
f. Add the following and write your answer in standard form:
(3.8 x 106) + (8.7 x 104)
Changing the indices to the same value gives the sum:
(380 x 104) + (8.7 x 104) = 388.7 x 104
= 3.887 x 106 in standard form
10

11. Worked examples
g. Subtract the following and write your answer in standard form:
(6.5 x 107) – (9.2 x 105)
Changing the indices to the same value gives the sum:
(650 x 105) – (9.2 x 105) = 640.8 x 105
= 6.408 x 107 in standard form
11

12. NEGATIVE INDICES AND SMALL NUMBERS
> A negative index is used when writing a number between 0
and 1 in standard form.
- 100 = 1 x 102
- 10 = 1 x 101
- 1 = 1 x 100
- 0.1 = 1 x 10-1
- 0.01 = 1 x 10-2
- 0.001 = 1 x 10-3
- 0.0001 = 1 x 10-4
- Note that A must still lie in the range 1 ≤ A < 10.
12

13. Worked examples
a. Write 0.0032 in standard form.
0.0032 = 3.2 x 10-3
b. Write 1.8 x 10-4 as an ordinary number.
1.8 x 10-4 = 1.8 ÷ 104
= 1.8 ÷ 10000
= 0.00018
13

14. Worked examples
c. Write the following numbers in order of magnitude, starting
from the largest.
3.6 x 10-3 5.2 x 10-5 1 x 10-2 8.35 x 10-2 6.08 x 10-8
Answer:
8.35 x 10-2 1 x 10-2 3.6 x 10-3 5.2 x 10-5 6.08 x 10-8
14

15. Fractional indices
= 41
= 4
> Therefore
but
> therefore
15
.
)
4
(
16 2
1
2
2
1
as
written
be
can
)
2
1
2
(
2
1
2
4
)
4
(


4
162
1

16
162
1

4
16 

16. Fractional Indices
In general:
16
n
n
a
a 
1
m
n
n m
n
m
a
or
a
a )
(
)
(


17. Worked examples:
a. Evaluate without the use of a calculator.
Alternatively:
= 21
= 2 = 2
17
4
4
1
16
16 
4
1
16
4 4
2

4
1
4
4
1
)
2
(
16 

18. Worked examples:
b. Evaluate without the use of a calculator.
Alternatively:
= 53
= 53 = 125
= 125
18
3
2
1
2
3
)
25
(
25 
2
3
25
3
)
25
(

2
3
2
2
3
)
5
(
25 

19. Worked examples
c. Solve 32x = 2
32 is 25 so
or
therefore
d. Solve 125x = 5
125 is 53 so
or
therefore
19
2
32
5

2
325
1

5
1

x
5
125
3

5
1253
1

3
1

x

20. 20
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