In today’s post I will be looking at how to Combine Powers. By a ‘Power’ I mean that little number you sometimes see at the top right of a number.

So – What does it mean?

3²  is another way of writing 3 x 3.  ‘Squares’ are quite familiar. But we can extend this idea

56  =  5 x 5 x 5 x 5 x 5 x 5 –  count the 5s – there are 6 of them!

And with that, you can probably how to see how any other ‘power’ calculation can be worked out, like 23¹⁵⁰ or 31²⁴⁵ – though you’ll understand if I don’t write those out in full!

Answers to ‘power’ sums can get very big!

By the way, sometimes you will see the word ‘index’ and that means the same thing. And sometimes its called ‘order’, which explains why it is an O in the acronym BODMAS.  If you are not sure what I mean by BODMAS, check here

How to Combine Powers

Once we understand how something is written in maths, the next step is to see how we can combine powers. How does this idea with things we already know?

For example – what does it mean if we write  6³  x 6⁴

The easiest way to see how to make sense of that is the first write this out in full

6 x 6 x 6    x   6 x 6 x 6 x 6  –    and now we have 7 6s  – all times together –  So we can write this as 6⁷

6³  x 6⁴ =  6⁷

Now, we don’t want to write things out in full every time, so lets look at what we have really done. Then we can see a short cut.

3 + 4 = 7!  So we can see a rule that might come from this. If you want to multiply power, just add the powers together.

Note that we can only do this if the bigger number is the same. We can’t add the powers and get any sensible answer if we try

5⁴ x  7³ – where the 5 and 7 are different. There is a way I would simplify that but I won’t look into that now.

Some special cases

One last thing for today,  I’d like to consider what 2¹, 2⁰ and 2^{-1 all mean.}

What does it mean to say ‘Multiply 2 and 2 and 2… -1 times! Doesn’t seem to make much sense, does it? But we can see how these things can mean something if we look again at our rule to combine powers.

So that’s not what we do, but we do want our rule of Index Adding to mean multiplying to still work.

So lets look at

2² x 2¹ = 2³

and 2² x 2⁰ = 2²

and 2¹ x 2^-1 =2⁰

The answers I’ve shown here have been worked out by the ‘adding powers’ rule – e.g. 2 + 1 = 3

The first one is the easiest to explain –
2 x 2    – how do we get that to 2 x 2 x 2 ?  By multiplying by 2 again!

so 2¹ = 2. That makes sense if you can get the sentence  2 multiplied together 1 time!

But what does it mean to say ‘2 Multiplied together 0 times’?

2² x 2⁰ = 2² – But what number doesn’t change others in multiply sums?

The only number that does that is 1…  so it can only make sense that 2⁰ -1

In fact, this is the rule for all numbers 5⁰ = 1 43⁰ = 1 – and so on.

This takes me to my special case, which complete the picture on how to use powers.

2¹ x 2^-1 =2⁰

Using the other two cases we can rewrite this as

2 x 2^-1 = 1

from that we can see 2^-1 = 1/2 – because thats the only number that completes this sum

We can extend that idea to say any ‘-‘ power  – just put the number on the bottom of a fraction with 1 on the top

So 14^-12 = 1/14¹²