A meme for math people

subtracting one from Exponent means halving (when the base is two):

2⁴ = 16 2³ = 8 2² = 4 2¹ = 2 2⁰ = 1

It's a simple continuation of the pattern and required for mathemarical rules to work.

This isn't strictly speaking a proof, but it did help me to accept it as it demonstrates the function that makes it 1.

2^3 = 2x2x2

2^2 = 2x2

(2^3)/(22) = (2x2x2)/(2x2) = 2

= 2^(3-2)

In general terms:

(x^a)/(xb) = x^(a-b)

If a and b are the same number this is x^0 and obviously (x^a)/(xa) is one because anything divided by itself is 1.

Hope that helps

Easiest explanation I can think of using the division law for exponents:

Since we can use any number for the initial fraction, as long as the denominator is the same as the numerator, any number to the zeroth power is equal to 1. In general terms, then, for any number, x:

0 is the neutral element for addition. This is why when we have a number then 0 + number = number (0 doesnt change the value in addition) and why 0 x number = 0 (if you add a number 0 times you will have 0). (Multiplication is adding one of the numbers to itself the number of times designated by the second number)

The same way 1 is the neutral element for multiplication. This is why when you have some number then 1 * number = number. This is also why number^0 = 1 (if you never multiply by a number you are left with the neutral element. It would be weird if powering by 0 left you with 0 for example because of how negative powers work)

This is the level 1 answer.

The level 0 answer is that it is this way because all of mathematics is a construct designed to ease problem solving and all people collectively agreed that doing it this way is way more useful (because it is)

Choose which one you want

You can think of 1 as the "empty product" (or the "neutral element of multiplication" if you want to be fancy). 2^x means you have x factors of 2. If you have 0 factors, you have the "empty product"

I see other people have posted good explanations, but I think the simplest explanation has to do with how you break down numbers. Lets take a number, say, 124. We can rewrite it as 100 + 20 + 4 and we can rewrite that as 1 * 10^2 + 2 * 10^1 + 4 * 10^0 and I think you can see why anything raised to the 0th power has to equal 1. Numbers and math wouldn't work if it didn't.

I like to think of it this way:
2^3 is the same as 2 x 2 x 2.
But you can arbitrarily multiply by as many 1s as you want because 1 has the identity property for multiplication.
So we can also write 2^3 as 2 x 2 x 2 x 1 x 1.
2^2 as 2 x 2 x 1 x 1.
2^1 as 2 x 1 x 1.
2^0 as 1 x 1 or just 1.

Multiplying a number by another number is the same as adding a number to itself that many times. And 0 is has the identity property for addition, so similarly:
2 x 3 = 2 + 2 + 2 + 0 + 0
2 x 2 = 2 + 2 + 0 + 0
2 x 1 = 2 + 0 + 0
2 x 0 = 0 + 0

Its not the same. And theres proof, why.

In addition to the explanation others have mentioned, here it is in graph form. See the where the graph of 2^x intersects the y axis (when x=0):

https://people.richland.edu/james/lecture/m116/logs/exponential.html

This also has some additional verbal explanations:

http://scienceline.ucsb.edu/getkey.php?key=2626

The simplest way I think of it is by the properties of exponentials:

2^3 / 2^2 = (2 * 2 * 2) / (2 * 2) = 2 = 2^(3-2)

Dividing two exponentials with the same base (in this case 2) is the same as that same base (2) to the power of the difference between the exponent in the numerator minus the exponent in the denominator (3 and 2 in this case).

Now lets make both exponents the same:

2^3 / 2^3 = 8/8 = 1

2^3 / 2^3 = 2^(3-3) = 2^0 = 1

2^0 isn't multiplying by zero. Considering this law: 2^a / 2^b = 2^(a-b)
it's obvious why 2^0 = 1
If a=b you're dividing by the same number resulting in 1.

Unfortunately, I cannot explain/prove the first law though.

This dude is great at explaining math, including this: https://yewtu.be/watch?v=r0_mi8ngNnM

Thanks, I couldn't even tell what the image was about math. I thought a dirty joke was hidden somewhere involving the 0. Didn't realize it was small and floating above on the right so people would immediately realize it's a power lol. Many people hide clever things but I always approach them in the wrong way lol.

Well looks like some people already answered your question but let me show you quick proof video that may help you understand how powers work: https://youtu.be/kPTp82EGjv8?feature=shared