ACCESS NUMBER
Sheet 10
INDICES AND STANDARD FORM
INDICES OR POWERS
A power, or an index, is used when we want to multiply a number by itself several times. It
enables us to write a product of numbers very compactly. The plural of index is indices.
There are a number of rules, or laws, which can be used to simplify expressions involving
indices.
We write the expression
Examples
5 × 5 × 5 × 5 as 54
a Using the first rule we can write
83 × 84 = 83+4 = 87
We read this as ‘five to the power four’.
b Using the first rule we can write
a4 × a7 = a4+7 = a11
Similarly a × a × a = a3
We read this as ‘a to the power three’ or
‘a cubed’.
You could verify the first result by
evaluating both sides separately.
In the expression 54, the index is 4 and
the number 5 is called the base. Your
calculator will probably have a button to
evaluate powers of numbers. It may be
marked xy or yx. Check this, and then use
your calculator to verify that 54 = 625
and 137 = 62748517
EXERCISE 2
In each case choose use the first law to
simplify the expression:
1 53 × 513
2 813 x 85
3 x6 × x5
EXERCISE 1
Work out the value of:
Second rule
1 43
am ÷ an = am−n
4 ( −2)3
5 ( −3)2
2 55
6 (-
3 26
4 a3 x a4
5 y7 x y3
6 x8 x x7
When expressions with the same base are
divided, the powers are subtracted.
1 2
)
2
Examples
The rules of indices
We can write
95 ÷ 93 = 95− 3 = 92
To manipulate expressions involving
indices we use rules, sometimes known as
the laws of indices.
and similarly
a7 ÷ a4 = a 7− 4 = a3
EXERCISE 3
First rule
m
n
In each case use the second law to
simplify the expression:
m+n
a ×a =a
When expressions with the same base are
multiplied, the powers are added.
1 513 ÷ 53
2 813 ÷ 85
3 x6 ÷ x5
1
4 a3 ÷ a4
5 y7 ÷ y3
6 x 8 ÷ x7
Third rule
Standard Index Form
When one power is raised to another
power we multiply the powers:
Standard form is a way of writing down
very large or very small numbers easily.
10³ = 1000, so 4 × 10³ = 4000 . So 4000
can be written as 4 × 10³ . This idea can
be used to write even larger numbers
down easily in standard form.
(am)n = am x n
Example
(35)2 = 35× 2 = 310
EXERCISE 4
In each case choose an appropriate law to
simplify the expression:
1 (53)12
2 (812)5
3 (x6)5
The rules when writing a number in
standard form is that first you write
down a number between 1 and 10, then you
write × 10 (to the power of a number).
4 (a3)4
5 (y7)3
6 (x8 )7
A quick rule for finding the index is to
count the number of places you need to
move the decimal point until it is just
after the first (non-zero) figure. (If you
move to the right, the index will be
negative.)
Negative Indices
Just by using the rules above we start to
come across more of our questions:
32 ÷ 34 = 32-4 = 3-2
But what does that mean? Let's write the
calculation without indices:
3×3
=
3×3×3×3
1
Examples
3x3
The two threes on top cancel two of the
threes on the bottom, so we can write
as
1
32
.
Write 81 900 000 000 000 in standard
form:
1
3x3
81 900 000 000 000 = 8.19 × 1013
It’s 1013 because the decimal point has
been moved 13 places to the left to get
the number to be 8.19
So negative indices give us one over our
number:
1
1
3-2 = 2 =
3
9
5-4 =
(-3)-5 =
1
54
=
1
3638 = 3.638 x 103
625
1
(−3)5
=
1
243
One over any number is called the
reciprocal of the number.
EXERCISE 6
EXERCISE 5
1
2
3
4
5
Write these numbers in standard form:
Work out the following, leaving your
answers as a fraction:
1 2-2
4 6-2
2 3-3
5 10-5
3 4-4
6 5-3
2
248
6800
781
563
7800
6
7
8
9
10
842
5200
343
230
4560
Small numbers can also be written in
standard form. However, instead of the
index being positive, it will be negative.
On a calculator, you usually enter a
number in standard form as follows:
Type in the first number (the one
between 1 and 10).
Examples
Press EXP or x10x .
Write 0.000 001 2 in standard form:
0.000 001 2 = 1.2 × 10-6
Type in the power to which the 10 is
risen.
-6
It’s 10 because the decimal point has
been moved 6 places to the right to get
the number to be 1.2
EXERCISE 9
0.0075 = 7.5 x 10-3
Using the EXP or x10x button on a
calculator, evaluate the following:
EXERCISE 7
Write these numbers in standard form:
1 0.6
6 0.56
2 0.2
7 0.054
3 0.09
8 0.061
4 0.05
9 0.1
5 0.45
10 0.087
5.2 x 104 = 52000
(move the decimal point 4 places to the
right)
x
2 x 102
2
8 x 103

2 x 102
3
5.3 x 103
+
8.2 x 103
4
4.5 x 10-3
-
5 x 10-3
5
4.6 x 103
x
5.9 x 10-1
7 The population of the USA in 2008
was 2.55 x 108, of whom 1/19 were over
75 years old. How many were over 75?
6.11 x 10-3 = 0.00611
(move the decimal point 3 places to the
left)
8 The capacity of a computer is 40
megabytes, where 1 megabyte is 1000
kilobytes. 1 kilobyte is 1.024 x 103
bytes. Express the capacity of the
computer in bytes in standard form.
EXERCISE 8
Evaluate these numbers:
5.3 x 103
3.4 x 102
6.9 x 105
9.2 x 10-1
9.43 x 10-1
3 x 102
6 In Peter’s blood there are 4.7 x 1012
blood cells per litre. Peter’s body
contains 4.9 litres of blood. How many
red blood cells are there in Peter’s
body?
It is important to be able to work out the
value of standard form numbers.
1
2
3
4
5
1
6 3.11 x 10-2
7 4.7 x 100
8 5.66 x 10-1
9 4.505 x 10-1
10 3.2 x 106
9 The number of people attending a
health centre is 132000 during a year:
a Give this number in standard form.
b Find the average number of people
attending on each day
3
4