Index Laws (Part I)

In this article, we’re going to be unpacking the index law for multiplying terms. This article is mostly aimed at students who are struggling to understand the index laws, but I wanted to begin with a motivation for this series of posts on the index laws, as I feel other educators may benefit from my experience.

Motivation

Index laws are an important part of mathematics, yet, early on, students are usually taught a very rote learney way of applying them. Students are usually introduced to the index laws through an algebraic form, before being exposed to heaps of examples.

For example, in this post we’ll be looking in-depth at the index law for multiplication. Students will usually be taught the law in algebraic form as follows:

\[a^m \times a^n = a^{m+n}\tag{Index Law I}\]

Then they’ll be shown some simple examples to cement this law in memory, such as:

\[x^3 \times x^7 = x^{10}\]

Now, I’m not suggesting that teaching students this way is inherently bad! In fact, when teaching students, this is the place I start off with. However, troubles arise when this is where the teaching stops. Just knowing index law I does not mean a student has an intuitive understanding of the underlying principle.

This issue was emphasised to me recently, when a student I tutor was confused by a question in a form similar to:

\[10x^3 \times 15x^7\]

The student incorrectly though that the answer to the question was:

\[10x^3 \times 15x^7 = 25x^{10}\]

This student was baffled as to why doing 10 + 15 to get 25 was incorrect here! The confusion lied with how in index law I, we are going from the left hand side terms being multiplied, to the right hand side where the indices are being added. On discussing with the student, there was a general confusion as to why we were allowed to add the indices, but we weren’t allowed to add the coefficients.

Developing a Conceptual Understanding

Now we are going to develop a conceptual understanding about what is going on under the hood of this law. And the law is actually quite easy to understand from a conceptual point of view.

Mainly, we want to unpack why it is totally legitimate to go from:

\[a^m \times a^n \]

To this:

\[a^{m+n}\]

The first thing we have to remember, is what does “a to the power of m” actually mean? Well, quite simple, it means that we are multiplying “a” by itself “m” times. In a similar fashion, “a to the power of n” means we are multiplying “a” by itself “n” times. Now, this is a bit abstract, especially since “a” could be anything, and so could m and n. So let’s start with an example! Let’s show that the following statment makes sense:

\[2^{2} \times 2^{3} = 2^5\]

Now, obviously you could just chuck it into your calculator, but let’s not do that…

First, we consider the right hand side:

\[\color{red}{2^{2}} \times \color{green}{2^{3}}\]

We know that “2 squared” simple means “2 times 2”. This gives us:

\[\color{red}{2 \times 2} \times \color{green}{2^{3}}\]

We also know that “2 cubed” means “2 times 2 times 2”. This gives us:

\[\color{red}{2 \times 2} \times \color{green}{2 \times 2 \times 2}\]

Now, it should be clear to see why “2 to the power of 5” is the answer. We are multiplying 2 by itself 5 times!

\[\color{red}{2 \times 2} \times \color{green}{2 \times 2 \times 2} = 2^5\]

Now, there shouldn’t be anything revolutionary about what I’ve just shown you here! However, by actually following through the expansion of the powers, it becomes clear that even though you may get confused about multiplication vs addition, what is ultimately happening here is just multiplication.

Taking This a Step Further

Now, let’s revisit the question (with colour coding) which stumped my student:

\[\color{red}{10x^3} \times \color{green}{15x^7}\]

Understanding conceptually what happens, this question should become a lot easier!

First, let’s expand the powers:

\[\color{red}{10 \times x \times x \times x} \times \color{green}{15 \times x \times x \times x \times x \times x \times x \times x}\]

Now, we know that multiplication is commutative (this is just a fancy way of saying the order we multiply doesn’t matter). Then, we can rewrite above as:

\[\color{red}{ 10 \times }\color{green}{ 15 \times }\color{red}{x \times x \times x} \times \color{green}{ x \times x \times x \times x \times x \times x \times x}\]

We can then work out was 10 times 15 is, and we also know that we are multiplying “x” by itself 10 times, giving us:

\[150 \times x^{10} = 150x^{10}\]

Conclusion

Through this post, we’ve developed a conceptual understanding as to why the index law for multiplication is valid. Hopefully this clears up any confusion as to why we are adding the indices, and how even though it appears we are doing addition, we are still ultimately doing multiplication.

In the next part of this series, we’ll look at the index law for division! Stay tuned!

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