What Exactly Is Zero Raised To The Power Zero?

An image asking the question: What is zero raised to the power zero?

Zero raised to the power zero is not something you come across in day-to-day life. When I say that, I mean both in real-life applications as well as in mathematics. So why does this topic interest us today? In a recent article that I wrote, I explained that any number raised to the power zero leads to one, except for zero. That is, x^0 = 1, except for x=0, where x is a positive number (for negative numbers, the result is -1 because -x^0 = (-1)*(x^0)). If you are curious enough about this topic, you would be itching to answer the question that follows from it: How come zero is an exception here?

Well, that is exactly the question that I am trying to tackle in this article. To start, I’ll first split the problem into understanding what happens when zero is the exponent with a non-zero base (x^0) and when zero is the base with a positive exponent (0^x). I’ll then combine the knowledge to attack the problem of 0^0 from different angles. At the end of this article, you will be able to appreciate the level of intricacy involved with this problem, and why exactly 0^0 is an exception to the general behaviour we have established so far.

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What is X^0, where x is a non-zero number?

In my previous article, I explained how any non-zero number that is raised to the power zero is actually divided by itself and results in 1. To appreciate this point of view, let us revisit the division rule of exponentiation. The division rule says that whenever we have a base raised to an exponent divided by the same base raised to another exponent, we just need to subtract the exponent of the denominator base from the exponent of the numerator base. It is perhaps easier to look at how this works via worked-out examples:

Explaining zero raised to the power zero using division rule:
(x^a)/(x^b)=x^(a-b)
Example: 2^5/2^3=2^(5-3)=2^2=4
2^6/2^3=2^(6-3)=2^3=8 — Math illustrated by the author

If we start from 1 and work our way towards 2^0 using the division rule of exponentiation, we prove our original point as follows:

Explaining zero raised to the power zero:
1 = 2^1/2=(2^1)/(2^1)=2^(1-1)=2^0
Therefore, 2^0=1 — Math illustrated by the author

So far, we’ve only revisited what I have covered in my previous article. Now, let us proceed towards the next part of the puzzle.

What is 0^x, where x is a non-zero number?

Let us now consider 0 as the base, and let x be any non-zero positive number. Whenever we apply the exponentiation function to a number, it results in a multiplicative series of the base occurring the exponent number of times. For example, consider 2^3. The exponentiation function is first resolved into a multiplicative series as follows: 2^3 = 2*2*2. The multiplication operation then gives the final result: 2*2*2 = 8. Therefore, 2^3 = 8. Similarly, 2^4 = 2*2*2*2 = 16.

If we now switch ‘2’ with ‘0’, we have no apparent reason why the logic would change. So, let’s just go ahead with it, and see what happens. Consider 0^4. Resolving it first into a multiplicative series, we get 0^4 = 0*0*0*0. We know that when zero is multiplied by itself any number of times, the result is zero. Therefore, 0^4=0. You will notice that the result is 0 for any positive number. Now suppose that we allow x to have negative values as well. When we consider negative exponents with a zero base such as 0^-1, we end up in a situation where we have to divide by zero, and this leads to an ‘undefined’ result. If you wish to understand this concept further, refer to my article on what really happens when you divide by zero. As it is now, we have sufficient knowledge to seek an answer to our main question.

What is Zero Raised to the Power Zero?

Let us take our findings so far, and try to use ‘0’ as both the base and the exponent. Based on applying the division rule of exponentiation, we figured out that any positive number raised to the power zero is nothing but the number divided by itself. Assuming this holds true for zero as well, we arrive at the following result:

Explaining zero raised to the power zero:
2^0 = 2^(1-1) = 2^1/2^1 = 2/2
Similarly,
0^0=0^(1-1)=0^1/0^1=0/0 — Math illustrated by the author

We have ended up dividing zero by zero (0/0). Unfortunately, 0/0 also leads to an ‘undefined’ result. It is important to note that the reason for an ‘undefined’ result, in this case, is completely different as compared to when a non-zero number is divided by zero. If you wish to understand the reasoning behind this problem further, refer to my article on what really happens when you divide zero by zero.

Based on what we have established so far, it appears that 0^0 leads to an undefined result. Shall we accept this as fact, and move on? We could. But let’s try solving the same problem using a different method.

Math Alert: If you are not keenly interested in technical mathematics, I suggest that you skip the following sections and head straight to the conclusion. You will miss the technical details, but will still be able to grasp the outcome of the analysis analogically.

Applying limits to Zero Raised to the Power Zero

To anyone interested in the technical details, welcome! If you have not heard of limits before, let me explain in simple terms. Using limits, we try and approximate zero using very small numbers close to zero. By doing so, we get a ‘feeling’ for what is going on around zero. The problem we are trying to solve here is 0^0. Let’s try and replace the zero with a small number close to zero. Since are not sure about how close we are going to go near zero, we represent this small number using x. At this point in mathematics, we call this form a function. There are different functions in mathematics that we can use to achieve 0^0. Let us consider 2 such functions which we’ve worked with before in this article: 0^x and x^0.

Considering the real number line, zero can be approached from the negative side, as well as the positive side.

The real number line with zero at the centre, +0.000001, +0.000002, etc., on the right and -0.000001, -0.000002, etc. on the left — Graphical Number Line – Illustrated by the author

The corresponding limits are be written as:

Explaining zero raised to the power zero:
lim x->0 (x^0). This can be split into two limits: lim x->0+ x^0 and lim x->0- x^0
lim x->0 0^x. This can be split into two limits: lim x->0+ 0^x and lim x->0- 0^x. — Math illustrated by the author

If you don’t get the mathematical convention here, don’t worry. This is not very important. You will still be able to follow the reasoning behind the analysis if you focus just on the numbers in the examples that follow.

Approximating Zero Using Limits

Let us consider two small numbers close to zero for our analysis further: +0.000001 and -0.000001. Let us now see what happens if we substitute these numbers in the first function: 0^x. We get the following:

Explaining zero raised to the power zero:
0^+0.000001 = 1
0^-0.000001 = 'undefined'
Based on this:
lim x->0+ 0^x = 0
lim x->0- 0^x = 'undefined' — Math illustrated by the author

0^-0.000001 is ‘undefined’ because we end up with the problem of dividing by zero again. Based on this, we could say that the answer approaches zero on the positive limit, and the answer is ‘undefined on the negative limit.

Let us now substitute the two numbers into the second function we considered: x^0. We get the following as the outcome:

Explaining zero raised to the power zero:
0.000001^0 = 1
-0.000001^0 = 1
Based on this:
lim x->0+ x^0 =1
lim x->0- x^0 = 1 — Math illustrated by the author

In this case, we see that the limits do indeed converge to 1. So far, we have had a variety of answers to 0^0 based on the context of the functions and the limits involved. As it turns out, depending upon the context of the problem we are trying to solve, 0^0 can take any value when limits are involved. As an example, consider the following result: