The pattern here is: Every time the exponent increments, the answer increases by a factor of 3. To get from 3^1 to to 3^2, you multiply 3 by 3 to get 9. Similarly, to get from 3^(-1) to 3^0, you multiply 1/3 by 3 to get 1.
This applies to exponentiation on any base, including zero (briefly checking a few examples, it seems to hold for all real numbers).
Wikipedia says 0^0 is commonly 1 in algebra and combinatorics, which I have more experience in. It is often undefined in computer science contexts. I was unaware of this, so thank you.
The more I learn, the more I realize there is no one universal math, only different rules which are helpful in different contexts. Thanks for bringing this point up.
A lot of people gave good responses. I think the best answer is that it depends on context. There are good reasons why it should be undefined, 1 or 0 depending on what you’re doing with it. Most people accept that it is 1, sometimes it makes more sense to be undefined or 0.
For an example for it to be 1, consider a combination lock on a briefcase. It has 4 digits, and each digit has 10 possible values. So there are 10^4 = 10,000 combinations. (Imagine the dials go from 0-9, so you have combos 0000-9999)
Now imagine the briefcase has no lock. How many ways can you can you arrange nothing? Just once.
The possible ways to arrange no objects in no possible ways is ∅ (empty set). So we have one possibility, empty set.
I still can’t wrap my head around why 0^0 = 1
If 0x0=0, and 0^0 is functionally identical to 0x0, why is it 1??
Anything to the power of 0 is 1. For instance:
3^(-2) = 1/9
3^(-1) = 1/3
3^0 = ?
3^1 = 3
3^2 = 9
The pattern here is: Every time the exponent increments, the answer increases by a factor of 3. To get from 3^1 to to 3^2, you multiply 3 by 3 to get 9. Similarly, to get from 3^(-1) to 3^0, you multiply 1/3 by 3 to get 1.
This applies to exponentiation on any base, including zero (briefly checking a few examples, it seems to hold for all real numbers).
0 to the power of anything is 0
0^3 = 0
0^2 = 0
0^1 = 0
0^0 = ?
Technically it’s undefined, but in most contexts you’re dealing with n^0 rather than 0^n, so it’s easier to just say it’s 1.
Wikipedia says 0^0 is commonly 1 in algebra and combinatorics, which I have more experience in. It is often undefined in computer science contexts. I was unaware of this, so thank you.
The more I learn, the more I realize there is no one universal math, only different rules which are helpful in different contexts. Thanks for bringing this point up.
A lot of people gave good responses. I think the best answer is that it depends on context. There are good reasons why it should be undefined, 1 or 0 depending on what you’re doing with it. Most people accept that it is 1, sometimes it makes more sense to be undefined or 0.
For an example for it to be 1, consider a combination lock on a briefcase. It has 4 digits, and each digit has 10 possible values. So there are 10^4 = 10,000 combinations. (Imagine the dials go from 0-9, so you have combos 0000-9999)
Now imagine the briefcase has no lock. How many ways can you can you arrange nothing? Just once.
The possible ways to arrange no objects in no possible ways is ∅ (empty set). So we have one possibility, empty set.
00 is undefined, but often it’s convenient to say it equals 1