Negative Indices

You can also have negative index numbers.  For instance, if I was given something like:


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we would say that the index is “negative five”.  So what do we do in this situation?  Well, you can change the negative index to a positive index if you invert the number.  So for this case:


I have inverted the 5 – which means writing down “one on five” as a fraction, and also changed the sign of the index from negative to positive – from ‘–2’ to ‘+2’.  You could, if you want, re-write it a little bit more by working out what the square of five is:


Powers raised to another power

The title of this section may sound a bit confusing, but once you see an example of what I’m talking about it’s pretty clear.  Say I have the following expression:


So we have the number ‘4’.  To the right and above it is a ‘2’ – so that part simply says, “four raised to the power two.”  However, above that ‘2’ there is a ‘3’ – so what does this mean?  Well, this means that the ‘2’ is also itself raised to the power ‘3’.

So we’ve got two parts raised to a power – the ‘4’ is raised to the power ‘2’, and the ‘2’ itself is raised to the power ‘3’.  Since we have more than one thing happening here, we need to know what order to do things in.  There’s a simple rule here – with powers you start from the bottom, and you work your way upwards.  So in this case, we start with the ‘4’ raised to the power ‘2’:


4 to the power 2 is 4 × 4 which we know equals ‘16’.  So we rewrite it as, “sixteen to the power three.”  By doing this we have simplified the expression into a more standard form.