What is 0 to the power 0?

Photo by Lum3n from pexels

We have always been amused by the hidden powers of the number “0”. It breaks division, completely changes multiplications, and a lot more!
The concept of what’s 0⁰ has been rolling since the start of indices.
Fortunately, we have an answer to it, that is 1! In this blog, we will explore why 0⁰ equals 1.

Indices refresher

Let’s quickly refresh our knowledge of indices before diving into the answer.
In maths, indices are the power or exponent which is raised to a number or a variable. It can be also said to be a repetitive multiplication.
Consider, 4². Here, 4 is the base number and 2 is the power.
4² = 4 * 4 = 16
There are a few laws that are to be kept in mind. A nice summary of those laws is given below:

Summary of the “Laws of Indices” image.
Image from Pinterest

So we know, anything⁰ = 1 and 0^anything = 0.
Let’s jumble them up and find what’s 0⁰!

Quest of finding what is 0⁰

Let’s begin! To find something unknown, we have to start with something that we know. So, let’s approach towards 0⁰ from 1¹!
I will be using an online calculator to save some time.

So, what do you see? It seems like as we approach 0, the answer is reducing, but the pace at which it is reducing has slowed down drastically. So, do you think the answer will be 0? Hold on, we are not done yet! Let’s continue.

Bingo! Totally not what you expected, right? It looks like now the answer is moving towards 1! Let’s dive even deeper!

Yes, yay! We are heading towards 1!

Since we are heading towards 1, 0⁰ is accepted to be 1!
So, that’s a good way to explain why 0⁰ = 1

Bonus!

Wanna impress someone? 👀
This equation of change explains what we just did:

All it says is, “as x approaches 0, x^x yields 1”.

Attribution of sources:
1. Eddie Woo’s Youtube Video
2. Wikipedia’s article

Thanks for reading, have a great day 👋

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