Anything multiplied by 0 is 0 right so why does x number raised to the power of 0 = 1? isnt it x^0 = x*0 (im turning grade 10 and i asked my teacher about this he told me its because its just what he was taught 💀)
In: Mathematics
Hiya. I think this question usually doesn’t get answered with a fully true answer. Hopefully this explanation makes some sense.
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When you raise a number X to a power Y, it means you multiply the number X by itself Y times, right? Like 5^3 means 5×5×5, we multiply 5 by itself 3 times.
So what is 5^0? Can we multiply 5 by itself zero times? That’s a nonsense statement, it doesn’t really mean anything.
So maybe we can clarify a bit: when Y is a natural number (1,2,3,4,…), raising X to the power of Y means we multiply X by itself Y times.
Now, we still want to get some kind of answer when we try 5^0, or even 5^-2, even though multiplying 5 by itself 0 or -2 times is nonsense. So, we see if there’s a pattern, and then we adjust the definition of raising to a power to fit that pattern.
The pattern we might notice is this:
If we go from 5^3 = 5×5×5 to 5^2 = 5×5, we divided by 5. If we go from 5^2 = 5×5 to 5^1 = 5, we again divided by 5.
So what happens if we continue the sequence down by just following this pattern and dividing by 5 over and over?
5^3 = 5×5×5 = 125
5^2 = 5×5 = 25 ( = 5^3 ÷ 5)
5^1 = 5 = 5 ( = 5^2 ÷ 5)
5^0 = ?? = 1 ( = 5^1 ÷ 5)
5^-1 = ?? = 0.2 ( = 5^0 ÷ 5)
5^-2 = ?? = 0.04 ( = 5^-1 ÷ 5)
While the original definition “you multiply 5 by itself that many times” stops making sense, the pattern of “you divide by 5 to get to the next one” can just keep giving us the next numbers in the sequence.
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At the end of the day, the answer to “why does 5^0 = 1” is “while it’s impossible to multiply 5 by itself zero times, if we just say it’s true that 5^0 = 1, it fits nicely with our pattern. So, we say it’s true”
x^0 is what’s known as an empty product, a product with no terms. It’s the product of 0 x’s. Mathematicians typically define empty products as 1, since that’s the result you get when you divide a product by all of its terms.
x5/X = x^(4)
x^(4)/X = X^(3)
X^(3)/X = X^(2)
X^(2) /X = X^(1)
X/X = 1 = x^(0)
1/X = X^(-1)
Anything multiplied by 0 is 0 (x * 0 = 0) – but why is that so? Because multiplying means addition multiple times, for example x*4 = x + x + x + x. Or you could say x*4 is first adding x three times, then adding x one more time: x*4 = x*3 + x*1. So far pretty obvious, so what if I wanted to say the same about 0? “Y*ou could say x*4 is first adding four times, then adding zero more times*”: x*4 = x*4 + x*0. For this last equivalence to be true, x*0 must be 0.
Now let’s rewrite exactly the same as above, but for power instead of multiplication:
Anything raised to the 0-th power is 1 (x^0 = 1) – but why is that so? Because power means multiplying multiple times, for example x^4 = x * x * x * x. Or you could say x^4 is multiplying by x three times, then multiplying by x one more time: x^4 = x^3 * x^1. So far pretty obvious, so what if I wanted to say the same about 0? “Y*ou could say x^4 is first multiplying four times, then multiplying zero more times*”: x^4 = x^4 * x^0. For this last equivalence to be true, x^0 must be 1.
The reasons in both cases are the same.
Terrance Howard mentioned this on Joe Rogan https://youtu.be/g197xdRZsW0?si=oXsqgfM9BXQ2-FuF
In the future if you don’t understand something in math, create an argument for what the thing should be before asking someone else. That’s the most important skill that even university students seem to struggle having the confidence to do.
x^0 doesn’t obviously mean anything. It isn’t x*0. Instead, it would be x multiplied with x zero times, which is paradoxical.
But x^m / x^n = x^m-n . So if m=n, you have 1 = x^n / x^n = x^0 . And it works like that with other formulas. So we decided, ok if we say x^0 = 1 , that makes sense, so let’s go with it.
ELI5:
You have a pack of 4 apples.
(4^1 = 4 apples)
You have a basket with 4 packs of 4 apples.
(4^2 = 4 x 4 = 16 apples)
You have a shopping cart of 4 baskets of 4 packs of 4 apples.
(4^3 = 4 x 4 x 4 = 64 apples)
You have zero packs of 4 apples.
(4^0 = 0 x 4 = 0 apples)
It’s a result of exponent rules. X^(2)*X^(3)=X^(2+3). Likewise X^(3)/x^(2) = x^(3-2)
For X^(0), we can say X^(0)=X^(y-y)=X^(y)/X^(y)=1.
For a practical example, we can say 2^(0) =2^(3-3)=2^(3)/2^(3)=8/8=1
X^(0)=1 is a natural consequence of exponent rules.
Not sure if anyone shared this yet, but you can think of powers as ways to arrange things.
You roll a 6 sided die, one time, there are six outcomes.
You roll a 6 sided die, five times, there are 6^5 outcomes.
If you don’t roll a die, there is one outcome. The non-action isn’t counted as zero, but it is the only possible outcome and is counted as one.
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