Okay, maybe I’m a bit slow but I said and thought something today that I’ve never thought about before.

We’re looking at indices & surds in my Year 11 class.  Today we were looking at the rules of indices: multiplying and dividing indices was “insultingly easy”.  Negative indices seemed to push it over the edge … some very confused faces.  We struggled with it and I think most people managed to get an instrumental understanding but not a relational one.  We’re working on that!

I was trying to explain that $6^{-3}$ was equal to $\frac{1}{6^3}$.

Now, in explaining this I said it was 1 divided by 6 three times.  Seems fine but where does the one come from?  We talked about reciprocals, we talked about the pattern that follows as we look at $6^3 6^2 6^1 6^0 6^{-1}$ … and we talked about a few more bits and pieces.

I then thought about $6^3$ could be thought of as 1 x 6 x 6 x 6.  This adds a bit of consistency to the system and made me think of an inaccuracy that I sometimes hear myself making or confirming.

“What does $6^3$ mean?”

“6 times by itself 3 times”

But does it? 6 times by itself three times could be 6 x (itself three times) or 6 x (6 x 6 x 6).  I suddenly thought surely this could be 1 times by 6 three times.  This helps with consistency in two ways.

Firstly, with our negative powers – we can say that it is 1 divided by 6 three times.  This has internal consistency and I like it.

Secondly, raising something to the power of 0 makes sense – 1 times by 6 zero times is obviously 1.

I’ve never seen or heard this anywhere before – am I wrong?  Does this break down somewhere that I’m not aware of?  Do you think this could be a helpful way to see it?

Also, is there a richer way to introduce this apart from pattern spotting?  Any ideas gladly received.

## 3 thoughts on “Thinking about indices …”

1. The ‘official’ explanation of powers is 1x whatever power you raise to;
To get them thinking about negative powers:
A negative exponent refers to the inverse of a number raised to a positive integer. Thus, a negative exponent should immediately suggest that the number in question has a value of less than one. For example, 2-3 = 1 / (2 * 2 * 2) = 1/8 = 0.125. Working with powers of 10:
10^-5 = 1/10^5 = .00001
10^5 = 1/10^-5 = 100000

(This is a good example because you can tie it back to scientific notation as an example.)

• Kevin Cunningham

I like that a lot … we talked about inverses and reciprocals but there was still a sense of confusion here. We’re coming back to it and I’m sure they’ll get there. I’ll make sure I make the link back to standard form (scientific notation) tomorrow … good link to make!

2. I like your new way of explaining it Kevin. I too would say that 6^3 is 6 times by itself 3 times and it’s never sat comfortably with me but I couldn’t think of a succint way to say it better. And they seemed to understand (although it surprises me how many times they still assume that 6^3, for example, is 18. No…”that would be 6+6+6, not 6x6x6″, I tell them. Any suggestions on how to beat that one?)