Video: How Should We Evaluate Zero to the Power of Zero?

In this video we look at several different options we have for how to define the expression zero to the power of zero, and think about the difference between definitions and results.

10:44

Video Transcript

In this video, we’re gonna think about how we can evaluate the expression zero to the power of zero. What’s its numerical value? There are a few strings of logic that lead to different answers, but which one is right?

First, let’s just recap what we mean by this power or exponent notation. For example, three to the power of two means that we write down two threes on the page and multiply them together. Three squared is equal to three times three, which is equal to nine. The three in this case is called the base, and the two is called the exponent.

Five to the power of four, or five with an exponent of four, means that we write down four fives on the page and multiply them all together. Five to the power of four is equal to five times five times five times five, which is equal to 625. And we can generalise that. 𝑥 to the power of 𝑦 means we write down 𝑦 𝑥s on the page and multiply them all together.

Great, so zero to the power of zero just means that we write down zero zero times and multiply them together. Oh dear, what does that even mean? Well, we could argue that whenever we multiply something by zero, the answer is zero. So however you define zero to the power of zero, it’s gonna involve multiplying by zero, and therefore the answer should be zero. Zero to the power of zero equals zero.

Well, that seems logical at first sight. But if we’re writing down zero zero times to multiply, then we’re not actually multiplying anything by zero because there are zero of them to multiply. Perhaps we could try to plot a graph of 𝑦 equals zero to the power of 𝑥 and see if there’s a logical value for 𝑦 when 𝑥 equals zero.

Well, it looks like we’ve got a straight line. And if we extrapolate that back to 𝑥 equals zero, it looks like we’ve got 𝑦 equals zero at that point. But that doesn’t prove anything though. There are plenty of functions that do something different for a particular 𝑥 value, for example, 𝑦 equals one over 𝑥.

In this case, it looks like the value of one over 𝑥 is approaching positive infinity as 𝑥 gets closer and closer to zero. So we might be tempted to suggest that one over zero is positive infinity, but wait! Approaching 𝑥 equals zero from the negative direction, we see the values of one over 𝑥 getting more and more negative, approaching negative infinity as 𝑥 gets closer to zero. So, is the value of one over zero equal to positive infinity or negative infinity?

Well, mathematicians say neither; they say one over zero is undefined. We need to get back to our graph of 𝑦 equals zero to the power of 𝑥 and try out some negative values. First though, let’s just recap what we mean by 𝑥 to the power of 𝑦 when 𝑦 is negative. Do we mean that we write out 𝑥 a negative number of times and multiply them together? Well, how could we even do that?

In fact, there’s a mathematical convention that turns out to have some other useful applications with exponents. Let’s start with three to the power of four, then think about three to the power of three, three to the power of two, and three to the power of one. As we increase the exponent by one each time, we’re multiplying the value of the expression by another three. Three to the power of one is three. We then add one to the exponent to get three to the power of two and multiply the expression by three to get three times three equals nine. So three to the power of two is nine, and so on.

When we add one to the exponent, we multiply the value of the expression by the base. And the other way, as we reduce the exponent by one each time, we’re dividing the value of the expression by three, the base. Now if we carry on applying that rule then, three to the power of one is three and three divided by three is one, which is three to the power of zero. Three to the power of zero is one because it means three divided by three, which gives us the result of one.

Well, under that convention, zero to the power of zero means zero divided by zero, which is undefined. But let’s carry on with our line of thinking using three as a base. So three to the power of zero is one. Let’s subtract one from the exponent again and divide the value of the expression by three again. Three to the power of zero is one, and one divided by three is a third. Then subtracting one from the exponent gives us three to the power of negative one. And if three to the power of negative one is a third, a third divided by three is a ninth, so three to the power of negative two is a ninth. And three to the power of negative three is one over 27, and so on.

Now we can see a pattern. Three to the power of one is three, and three to the power of negative one is one over three. Three to the power of two is nine, and three to the power of negative two is one over nine. Three to the power of three is 27, and three to the power of negative three is one over 27, and so on.

So, in general, 𝑥 to the negative 𝑦 is equal to one over 𝑥 to the power of 𝑦. And this convention is useful and consistent with the rest of the logic around exponents. Now we can get back to thinking about our tables of values of zero with negative exponents.

We saw that 𝑦 is zero for positive values of 𝑥. But oh dear, zero to the power of 𝑥 is undefined for all negative values of 𝑥. That doesn’t help in our quest to find out how to evaluate zero to the power of zero.

If we approach zero to the power of zero with smaller and smaller positive exponents, zero to the power of three, zero to the power of two, zero to the power of one, then it looks like zero to the power of zero should be zero. But if we approach it with negative exponents closer and closer to zero, zero to the power of negative three, zero to the power of negative two, zero to the power of negative one, and so on, then they’re all undefined.

It doesn’t seem obvious whether zero to the power of zero should be undefined or not, because zero isn’t a negative exponent. But from this direction, it doesn’t look like zero to the power of zero should be equal to zero.

Okay, let’s try another approach. Let’s look at 𝑦 equals 𝑥 to the power of zero. As we just saw, nine to the power of one is nine. So reducing the exponent by one means that we divide the value of the expression by the base, nine in this case, and nine divided by nine is one. So nine to the power of zero is one. Similarly, six to the power of zero is one, three to the power of zero is one, and it looks like zero to the power of zero ought to be one as well.

And if we have a look at negative exponents, again, negative nine to the one must be just negative nine. So reducing the exponent by one to zero, we must divide the expression by negative nine, and negative nine divided by negative nine is equal to one, so negative nine to the power of zero is one. Likewise, negative six to the power of zero is one, negative three to the power of zero is one, and so on.

Well, this looks pretty convincing. Thinking about 𝑦 equals 𝑥 to the power of zero, when 𝑥 is negative, as it approaches zero, 𝑥 to the power of zero is consistently one. As 𝑥 is positive and approaching zero, it’s also consistently one. Surely zero to the power of zero must also be one.

The thing is, having a value close to zero isn’t the same as having a value of exactly zero, so we haven’t proved anything. It’s a bit like a raffle in which each of 10 people gets a ticket with a number from one to 10 on it. Then a number in that range is chosen at random, and the holder of the ticket with that number wins a fabulous prize.

Let’s say that ticket number five is the winner. Using the logic that we used with 𝑦 equals 𝑥 to the power of zero, we could say that ticket numbers one, two, and three and four are all losing tickets and tickets 10, nine, eight, seven, and six are too. So whichever way you approach ticket number five, you’re getting losers, so ticket number five must also be a loser.

We’ve overlooked the fact that there’s something unique and spectacularly different about ticket number five: it’s the winner. So we’ve come up with three possibilities for the value of zero to the power of zero. It could be zero, or undefined, or one. But there doesn’t seem to be a consistent and entirely logical killer argument for it to be any one of those in particular.

Well, it turns out that, for mathematicians, it’s most useful if we define zero to the power of zero as being equal to one, so that’s what we’ve done. There isn’t an answer that you can prove. But choosing to use a value of one for the expression zero to the power of zero doesn’t break any other rules, and it’s convenient.

Sometimes we define things that seem to make sense, and sometimes we use logic and build provable theorems from our definitions. This is one of those occasions where we’ve just made a group decision to define something in a certain way because it looks useful.

A reasonable, simple example of this sort of thing is when we define a straight line as being the shortest one-dimensional path between two points. Technically, it then extends infinitely beyond the two points in either direction. Now we can pick a point on the line and define the angle measure from the line in one direction to the line in the other direction around that point as being 180 degrees.

We can now derive and prove a number of rules using straight lines. For example, when two lines intersect at a point, then vertically opposite angles must be equal. Then if we introduce a parallel line, another definition, we can call these angles alternate angles and prove that they must be equal in measure. We can then define a polygon as a two-dimensional plane enclosed shape with three or more straight edges and then a triangle as a polygon with specifically three edges.

Now we can use the definitions and proofs we have so far to prove that the sum of the interior angles in a triangle is 180 degrees. Some parts of mathematics are definitions, and some are logical developments of those definitions.

Back to our zero to the power of zero definition then, there’s a thing called binomial theorem, which, amongst other things, helps us to quickly multiply out expressions like 𝑎 plus 𝑏, all to the power of 𝑛. For example, to expand 𝑥 plus three all to the power of six takes quite a lot of working out if you do it longhand. But you can more or less write it straight down if you know the binomial theorem.

Now if we had an expression like zero plus 𝑥 all to the power of six, then clearly zero plus 𝑥 is just 𝑥, so this is just 𝑥 to the power of six. And the binomial expansion of that would be this. And zero to the power of one, zero to the power of two, and so on are all zero, so these terms are all multiplied by zero and become zero. But if zero to the power of zero was also zero, then that first term here would become zero too, and we’d be saying that zero plus 𝑥 all to the power of six is equal to zero, and that just wouldn’t be right. It’s much easier to define zero to the power of zero as being equal to one, and then this works just fine.

Now, there are other ways around this problem. And this isn’t the only reason for choosing to define zero to the power of zero as being equal to one, but it is an example. The whole of mathematics starts with seemingly sensible definitions known as axioms. And from these, the rest of mathematics is built. One of the things that’s been agreed along the way is that it would be useful if zero to the power of zero was defined as being equal to one.

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