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In mathematics, we often need to find a shorthand way of representing information or data. Nowhere is this need more obvious than when we wish to represent something like the product of 2 multiplied by itself 2, 6, 10, 15 or even 20 times.
Instead of writing 2 × 2 × 2 × 2 × 2 × 2, we write 26. This is read (and said) as ‘2 to the power 6?; 6 is the index of the power. In general, this means that
where n is called the index.
Both a and n can be either positive or negative numbers; a−n can also be written as 
.
From this simple definition, we can now go on to look at power notation, or index notation as it is often called, in more detail. We will establish the five rules that are used with this notation.
Rule 1 
a3 × a 2 = (a × a × a) × (a × a) = a5
Thus a3 × a2 = a(3 + 2) = a5
This rule works for both positive and negative indices.
View larger image Thus a−3 × a5 = a(−3 + 5) = a2
Rule 2 
Thus a5 ÷ a2 = a(5 − 2) = a3
This rule works for positive and negative indices but you need to take care when dividing by a negative power.
View larger image Thus a2 ÷ a−6 = a(2 − (−6)) = a8
Rules 1 and 2 only work if the powers involve the same factor, for example, you can't simplify a3 × b2 or a7 − b5.
Rule 3 
This is because
View larger image and, using the rule for division (Rule 2)
Remember a = a1
Note that we can now explain why 
a−n can be written as a(0 − n)
and, using Rule 2, this is the same as a ÷ an
which, using Rule 3, is the same as
In the same way, we can show that 
can be written as 
so, using Rule 2 and Rule 3
Rule 4 
We know that a2 = a × a
So, (a3)2 = a3 × a3 = a(3 + 3) = a6 = a(3 × 2)
This rule works for positive and negative indices.
For example, (a−4)3 = a−12
Rule 5 
Since a1/2 squared is equal to a, a1/2 is the square root of a and is sometimes written as 
. a1/3 cubed is equal to a so a1/3 is the cube root of a and can be written as 
. a1/4 to the power of 4 is equal to a which means that a1/4 is the fourth root of a, written as 
.
So it follows that
a1/n is the nth root of a and can be written as 
.
If we use Rule 4 to write ap/q as (ap)1/q or (a1/q)p, you should now be able to see that ap/q is the qth root of ap which is written as 
.
Rules 1, 2, 4 and 5 all work with fractional indices.
Simplify each of the following:
(a) 
(b) 
(c) 
(d) 
(e) 
(f) 
(g) 
(h) 
(a) 
(b) 
* you might like to check the answers to (a) and (b) using a calculator.
(c) 
(d) 
(e) 
(f) 
(g) 
(h) 
Simplify:
(a)
(b)
(c)
(a)
(b)
(c)
Although you may arrive at the answer to an activity in a different way from that given in this toolkit (because the rules of indices can often be applied in a different order), you should always get the same final answer.
All the examples we have discussed in this section have a simple answer but this will not always be the case, particularly where the calculation relates to a situation modelled on the real world.
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