Indices and powers

Sometimes you multiply identical numbers together.  For instance, I could multiply together 4 and 4 like this:

                                                             

Now this is a pretty easy thing to write down on paper isn’t it?  But what about if I multiplied the same number by itself a whole lot of times, like this:

                                          

This takes a lot more time to write and it’s easy to get lost as you read through all the numbers and multiplications.  Luckily, there’s an easy way in mathematics to write something like this.  What you need to do is use an index number.  Index numbers are what you write above the right hand side of a number to tell you how many times that number is multiplied by itself. 

Now say we wanted to represent  using an index number.  What you need to do is count up the number of ‘4’s in the expression – there are two of them.  Now all you need to do is write a 4, with an index number of ‘2’:

                                                               

What about for the ?  Well, all you need to do is count the number of ‘5’s – there are 10 of them, and write this ‘10’ as the index number:

                                                               

Easy!  Now, what about going back the other way?  Well, say we came across something like:

                                                              

To write this out the long way, you look at the value of the index number – it tells you how many ‘14’s there are in the multiplication.  In this case, the index number is ‘4’ – so there are four lots of ‘14’ in the multiplication.  So writing it the long way it should look like:

                                                     

What about something like:

                                                            

This expression has more than one index number in it – it contains indices.  ‘Indices’ is the plural form (meaning more than one) of the term ‘index number’.

Some people prefer to talk about something like  as “four to the power three”, so don’t get confused if you hear someone describing it that way.  Also, you aren’t limited to using indices only with numbers – you can use them with algebraic variables as well, for instance:

                                                             

This is really