Exponents

In our first encounter with exponents, we are taught that exponentiation is notation that simply means “multiply the number by itself some number of times.” For instance, 3⁵ means the same thing as 3×3×3×3×3. For most people, this is a perfectly adequate definition. To a mathematician, it begs for exploration.

Working only with integers, there are some interesting properties that can be worked out. Some of these are listed below.

For instance, consider the product rule. It states that if we are given something a product of two exponential expressions with the same base, we can add the exponents together to get a new exponential expression which is equivalent to the original expression. Intuitively, this should make sense. Consider the following:

\(\left(x^2\right)\left(x^3\right) = (x\cdot x)(x\cdot x\cdot x) = x\cdot x\cdot x\cdot x\cdot x = x^5 = x^{2+3}\).

While this does not constitute proof in a mathematical sense, the argument seems pretty convincing to me. If we are multiplying two exponential expressions, then we can write each of those expressions as a product, rather than an exponential, then remove the parentheses using the associative property of multiplication. Once that is done, we are left with a string of xs which are multiplied together, which fits with our notion of what exponents mean.

All three of the above properties can be worked out in a similar fashion. The curious mathematician would begin by expanding out all of the multiplication, then grouping or ungrouping terms as is appropriate, eventually geting from the statement on the left hand side of the equal sign to the statement on the right hand side. I would invite the reader to give it a try, and confirm that the properties really do seem to hold.

All three of the properties above are related to multiplication of exponential expressions. This makes sense, as exponentiation can be thought of as a kind of extended multiplication. To a mathematician, however, only multiplying gets boring pretty quickly. Because multiplication and division are somehow related (division can be thought of as the inverse of multiplication—in a sense, division “undoes” multiplication), the mathematician might next ask, “Okay, so what happens when I divide?” He or she might try something like the following:

\({x^7\over x^4} = {x\cdot x\cdot x\cdot x\cdot x\cdot x\cdot x\over x\cdot x\cdot x\cdot x} = x^3 = x^{4-3}\).

This is an interesting result. It looks as though when exponential expressions with the same base are divided, we can subtract the exponents from each other. This gives rise to the following rule:

The quotient rule actually gives rise to another couple of interesting properties. If we take m and n to be equal to each other, we get a rule that states

Or we could pick \(m=0\) and some bigger number for n. This gives us a really interesting result:

Suddenly, we know how to work with not just positive exponents, but negative exponents, as well. Just by playing around with exponential expressions, we discovered some rules and regularities for exponents. From there, we discovered that our concept of an exponent could be made much more general than our original idea of repeated multiplication.

Suppose, for instance, I asked you, out of the blue, “What does it mean to multiply a number by itself negative three times?” The question is nonsense. You would have no idea how to even begin approaching it. However, while playing around with exponents, we figured out that negative exponents turn into division problems! We may not be able to answer the original question, exactly, but we can come up with an answer that makes sense!

This actually brings me to the topic that I am in the midst of teaching in my algebra classes. Rather than simply introducing some new notation, I like to motivate the discussion by asking a question that seems really silly at the outset: what does it mean to multiply a number by itself 1/2 times?

Right off, this question seems really silly. It doesn’t make any sense. But let me ask it another way: what is \(x^{1/2}\)? We understand exponents to mean repeated multiplication, so an exponent of 1/2 seems a little strange.

Instead of trying to answer the original question, as posed, it might make more sense to attempt to understand the question in terms of notation. We would really like \(x^{1/2}\) to have the same properties of our other exponential expressions. That is, we would like it to obey the product rule, the power rule, and the other four rules outlined above (as well as some other rules).

So here’s an idea: let’s just assume that it does follow the rules. In a sense, mathematics is just a game, and we can make up any rules that we like, so why don’t we just declare that \(x^{1/2}\) follows all of these rules?

For instance, we can use the product rule to find out that \(x^2\cdot x^{1/2} = x^{5/2}\), or we can use the product rule to determine that \(\left(x^1/2\right)^3 = x^{3/2}\). Under the assumption that \(x^{1/2}\) behaves the way we want it to, these exponential expressions make sense. Unfortunately, this result is not that enlightening. In fact, it might even be more confusing. Now we have to think about not just \(x^{1/2}\), but also \(x^m\) where m is any rational number!

Perhaps we need to try some other numbers, and see what happens. For instance, note that

\(x^{1/2}\cdot x^{1/2} = x^{(1/2)+(1/2)} = x^{1} = x\).

Now we seem to be getting somewhere! We took our mystery number \(x^{1/2}\), squared it, and got x. Put another way, we have

\(\left(x^{1/2}\right)^2 = x\).²Note that we could have gotten this same result using the power rule. This is a nice indication that our definition still makes sense. If we assume that fractional exponents behave as we want them to, then we should get the same result if we do the same thing in two different ways.

Instead of trying to think about repeated multiplication, let’s have a look at what is going on here. We have a magic number called \(x^{1/2}\) which, when we square it, gives us x. This should sound familiar to us. In elementary school, you were probably introduced to something called the square root of a number. The square root is a number that, when we square it, gives us our original number. We might denote that using a radical sign (\(\sqrt{}\)).

It looks like \(x^{1/2}\) has the properties of a square root. Specifically, notice that if we replace \(x^{1/2}\) with \(\sqrt{x}\), we get the results that we expect:

\(\left(\sqrt{x}\right)^2 = x\).

What is really interesting about this result is that it arose naturally when we asked what it meant to take a number to the one half power. It might not have made sense to ask that question in the first place, but it turns out that the answer is something that we already know a little something about, namely square roots of numbers.

This leads me to a more general point. Mathematics is often taught as a bunch of steps, procedures, or algorithms that have to be memorized. In a lot of algebra classes, an instructor might simply give a definition. For instance, they might write

This is a perfectly valid definition. In fact, it is exactly the definition that we would eventually arrive at using the logical process begun above. However, this definition is sterile, and utterly removed from the process of mathematics. The definition did not pop from the head of Pythagoras fully formed—someone³Probably Gauss… had to ask the questions asked above, play with exponents, and figure out what happens and what it means.

Mathematics is a process, not a series of definitions that require memorization. I have students that just want me to write definitions on the board, so that they can memorize them and move on. I have other students that work hard to grapple with the process behind those definitions. I’ll give you three guesses as to which students enjoy my classes more, and which students do better on exams. ;)