Why does x⁰ = 1?

The short answer

When you multiply exponential expressions with the same base number, you add the exponents together:

2² * 2¹ = 2²⁺¹ = 2³

By that logic, you can prove that any non-zero number to the power of 0 has a value of 1:

2² = 2²⁺⁰ = 2² * 2⁰ = 2² * 1

The long answer

For anyone who hasn't taken a mathematics course in more than 10 years (🙋‍♀️ me), why don't we start with a refresher on exponents?

An exponent is a notation that represents the number of times a base number is multiplied by itself. In this example, 3 is the base number and 2 is the exponent.

3² = 3 * 3 = 9

To understand why any number to the power of 0 is always 1, let's try looking for a pattern in non-zero exponents.

2⁴ = 2 * 2 * 2 * 2 = 16
2³ = 2 * 2 * 2 = 8
2² = 2 * 2 = 4
2¹ = 2 = 2

Okay, let's stop there. Are you seeing a pattern?

Each time we step down in an exponent (i.e. from 2⁴ to 2³), the result is divided by the base power (2).

2⁴ = 16
2³ = 2⁴/2 = 16/2 = 8

If we carry this pattern all the way to 0, watch what happens.

2¹ = 2
2⁰ = 2¹/2 = 2/2 = 1

So 2⁰ is equal to 2¹ divided by 2 (the base number). This works for any number (well, almost any number, see the "🧠 Bonus brain points" section).

100¹ = 100
100⁰ = 100¹/100 = 100/100 = 1

There you have it, the wonders of math!

(By the way, ​this video explains this concept in a slightly different way​. If my explanation didn't make sense, perhaps theirs will!)

🧠 Bonus brain points

Why doesn't 0⁰ = 1?

Zero to the power of 0 is the only number that doesn't follow the rule above. That's because you can't divide anything by its base number, 0. Why is that?

Well, dividing is just the opposite of multiplying. So another way to write:

8 ÷ 2 = x

is

2 * x = 8

If you tried to replace 2 with 0 in that same multiplication formula, you run into a problem.

0 * x ≠ 8

Nothing multiplied by 0 can be anything but 0. So therefore, nothing divided by 0 can exist.

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Sources

Learneo, Inc. (n.d.). Explain with words and an example how any number raised to the zero power is 1?. CliffsNotes. https://www.cliffsnotes.com/cliffsnotes/subjects/math/explain-with-words-and-an-example-how-any-number-raised-to-the-zero-power-is-1

Mashup Math. (2015, September 21). WHAT IS AN EXPONENT IN MATH?. YouTube. https://www.youtube.com/watch?v=W3lPO3LrVuE

Simple Math. (2017, March 14). Why does "x to the zero power" equal 1? YouTube. https://www.youtube.com/watch?v=yiwAS3R-mG0

Simple Math. (2017, March 15). Why can’t you divide by zero? YouTube. https://www.youtube.com/watch?v=hv2dSnfTrxU