why does 0 to the power of 0 equal 1?
In: 97
0⁰ is undefined. That doesn’t mean it can’t have a value, it means there isn’t enough info to determine based on that alone. Let me give you an example. If you’re familiar with limits, this may be a bit tedious for you. So skip ahead as needed.
Let’s take the limit as x approaches 0 of 0^(x) this means we look at the value of the expression as x gets closer and closer to 0 (meaning the expression gets closer and closer to 0⁰) at x=0.1 the expression is 0^(0.1) which is 0 at x=0.01, the expression is 0^(0.01) which is also 0 and at x=0.0000001 it’s 0^(0.0000001) which is also 0. The limit is 0 because this pattern holds for any x arbitrarily close to 0. Therefore 0⁰ can equal 0
Let’s take the limit as x approaches 0 of x^(0) this means we look at the value of the expression as x gets closer and closer to 0 (meaning the expression gets closer and closer to 0⁰) at x=0.1 the expression is 0.1^(0) which is 1 at x=0.01, the expression is 0.01^(0) which is also 1 and at x=0.0000001 it’s 0.0000001^(0) which is also 1. The limit is 1 because this pattern holds for any x arbitrarily close to 0. Therefore 0⁰ can equal 1
By changing how we approach 0⁰, we can make it equal many different things.
What does it mean to take something to the power of zero?
Well, what does it mean to take something to a power *in general*?
a^n means multiply a by itself n times. So 2^3 is 2 x 2 x 2; 5^7 is 5 x 5 x 5 x 5 x 5 x 5 x 5.
What does a^0 mean, though? It means that you multiple something by itself…zero times? One way to interpret this is “What number, when multiplied by a once, yields a?”. The answer is 1. So mathematicians have decided on the rule that a^0 = 1. This is true regardless of what your a is. And this rule holds for 0 as well: 0^0 x 0 = 1 x 0 = 0, as expected.
This is the justification that lies behind the practice of 0^0 = 1. However, sometimes it’s also said that 0^0 is undefined; that is, mathematicians agree that 0^0 isn’t really a thing you can do.
The precise mathematical phrase 0^0 is not really applicable to real life; that means that the “true meaning” of the value is less important than whether the value is convenient (or easy or logical or whatever) to work with.
So, some mathematicians (mostly the ones that take weird things to weird powers a lot and want to be able to handle the edge cases with zero without having to write a special note at the bottom of their proofs) go with 1, some (mostly the ones who think it’s important to know precisely what statements can and cannot have meaning) go with undefined, because while 1 x 0 = 0, so does 2 x 0 and pi x 0 and -1,439,239.2384 x 0. And so it’s mostly a matter of usefulness and to some extent aesthetics.
Because 1 is the “neutral element” of multiplication.
Power a^b means you multiply a b-times with itself (a×a×a×…×a)
Now any result of that can be multiplied by 1 without changing it 1×(a×a×a×…×a)
Okay, now a and b are both zero.
So the equation is just 1, because the brackets don’t contain any terms you multiply with.
I figured I’d add a second comment answering a closely related question, but is more well defined. Why are other numbers (besides 0) raised to the power 0 equal to 1. In other words why is x⁰=1 for x≠0. Every time you increment the power by 1, you multiply by the base. x^(n+1)=x•x^n and as an example 2⁵=32=2(16)=2•2⁴ which makes sense because for integer powers, were taught the exponentiation is repeated multiplication. 2⁴=2•2•2•2 and multiplying by one more 2 is just 2⁵.
Symmetrically, every time you decrement the exponent, you divide by the base. x^(n-1)=x^(n)/x and as an example 3³=27=81/3=3⁴/3. And if you keep dividing by 3, you’ll eventually get to 3¹=3. Divide by 3 once more and you get 3⁰=3/3=1 boom. And you can even keep going, 3^(-1)=1/3. And 3^(-2)=1/9=1/(3•3)
Lots of correct answers here. 0^0 is undefined.
However, an interesting way to look at equations is to analyze what happens to functions when you play with the numbers. If you load up a graphing calculator, you can plot functions that include 0^0. The simplest function to understand this is [x^x](https://www.wolframalpha.com/input?i=x%5Ex). The graph gets really close to 1 as x approaches 0, but the line never touches. It gets infinitely close, but it never touches.
For this reason, it is perfectly valid to say that the limit as x approaches zero is 1. The the exact value, however, is undefined. You can plug other functions into a graphing calculator that include 0^0, and you will find that all of them will have the same limit of 1, yet the line never touches the y axis.
Yes 0^0 is undefined (and sorry, don’t know how to do superscript on mobile so you get carat notation).
However, 0^0 = 1 is convenient in many situations. For example, imagine you’re doing a big, complex calculation on a computer. If you assume 0^0 is undefined, it will go and screw up your calculations.
However, often times, just letting it equal 1 is sufficient. This can be the case for example in some machine learning cases, amongst other things. Letting it be 1 still gives you solid, sensical results, so is it really a problem?
So the real question is, what are you using it for and why does it matter? If you are doing a highly theoretical mathematics proof, it’s probably best to consider it undefined. In some cases, maybe it’s best to consider it as 1. For the machine learning example, if you said it were 0^0 is undefined, or equal to 0, or equal to 50, or anything else, then you risk ruining a perfectly good algorithm.
Furthermore, perhaps there’s some wild scenario where saying 0^0 = 50 could make sense, and any other value is nonsensical. But most often, if you need a concrete value and not just undefined, 1 tends to have the properties that fit the best, even if it’s not mathematically precise.
My high school algebra teacher explained it in a stunningly simple way:
Raising a number to the nth power means multiplying it n times, and multiplying anything by 1 doesn’t change it.
So for example, 2^3 = 2 x 2 x 2, which is the same as 2 x 2 x 2 x 1.
If you take 2 to the power of 0 there are no 2s, there’s just the 1 by itself.
This is a strange question since in most contexts 0^0 is considered to be undefined. Short answer is there are a few formulas like the Taylor Series or the Binomial Distribution formula that would break in an edge case where you have 0^0 and to fix those cases, 0^0 would be 1.
It doesn’t. It’s undefined.
This doesn’t mean its value is mysterious or unknown, it means the expression “0 to the power 0” is nonsensical. The fact that you can write it down doesn’t make it valid.
The same is true of stuff like 1/0, because it means “the thing you multiply by 0 to get 1”. There is no such thing, so the expression itself is an error. And no, the answer isn’t “infinity” – infinity isn’t a number, it’s a useful shorthand which means all sorts of different things in different contexts.
The ELI5 answer is “beacuse mathematicians decide it was simplest if it did.” Any real answer is going to need calculus or more complex math to demonstrate
Here is how I think of it: x^0 = x^n * x^-n = x^n / x^n = 1 for all non zero cases because 0/0 is undefined. However we can use L’Hopital’s rule to determine it converges to a specific value. The first iteration of the rule transforms the equation into x*n / x*n, which is still undefined. The second itteration transforms it to n/n which is 1. So because the numerator and denominator are growing at the same rate, 0^n / 0^n = 1, therefore 0^0 = 1.
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