# A huge conflict: What is zero raised to the power of zero?

## What is zero raised to the power of zero? This is a question that asked more than 35 billion and 378 million times. And 98% of people hasn’t answered correctly.

First, what does 2⁵ mean? It means 2 times 2 times 2 times 2 times 2. In other words, multiply 2 by itself 5 times. Now, we can say 0⁰ means “multiply zero by itself 0 times”. Hmmm, that’s awkward.

Let’s go to different directions and find the other powers.

Once when we see an exponential equation such as 0⁹ = 0 , we will say “zero to the ninth power is zero”.

It looks like 0⁰ = 0. But 0 to the -5th power is 1 over 0 which is undefined and same with 0 to the -100th power. The negative exponents indicate 0⁰ should be undefined.

Let’s attack this from a different angle. Other numbers raised to 0 equal 1.

This pattern indicates that 0⁰ should also be 1. So, it looks like there isn’t certainly a particular accurate solution? Which is exact? Nonetheless depending on the situation, you work in one answer may be better than the others. The **best** explanation should be reliable, reduce needless complexity, and be beneficial.

Most theoreticians choose that in many cases, 1 is the finest definition for 0⁰. Let’s look at two reasons for this. ** a** raise to

**can be viewed as the number of sets of**

*b***elements that can be chosen from a set of**

*b***elements.**

*a*For example, 2¹ can be observed as the amount of sets of one element that can be chosen from the set of two elements.

And 0⁰ is the amount of sets of zero elements that can be selected from a set of zero elements. Which must be 1! So, 1 is the only definition reliable with this understanding of exponentiation.

In this perspective, any other definition would unnecessarily confound things. For another case where 0⁰= 1 is a beneficial definition, let’s look at the binomial statement.

As x = 0, this simplifies to 1 = 0⁰ • 1. In this item, the only explanation for 0⁰ that constructs the binomial statement correct is 1. Again 0⁰= 1 is the only definition that avoids needless complexity. Yet, depending on the sort of mathematics we are doing, 1 may not permanently be the finest definition.

For example let’s look at some limits. The limit of a function at point ** a** is the value of the function approaches as its input approaches

*a**. W*e’re involved in limits of the form 0⁰ when x = 0. A simple one is the limit of x⁰ as

**approaches 0. Since**

*x***= 1 at all other points, its limit at 0 is 1 as well. This seems to verify that 0⁰ = 1.**

*x⁰*Nonetheless there are other limits of the form 0⁰ with different values! The limit of 0 raise to x from the right is 0… And from the left it’s undefined. And other limits of the form 0⁰can be any value like this one which is *e**.*

*Th*ese conflicts are good reasons to call 0⁰ an “indeterminate form” or “undefined” when you’re dealing with limits. These are the only definitions that are consistent with the way we define limits.

So what is 0⁰? It depends! Often, 1 is the best answer. However, when dealing with limits, “undefined” or “indeterminate form” os more sensible. Depending on the type of math we are doing, even definitions and conventions can change!