$0^0cdot0^x=1cdot0^0$, so $0^0=1$$0^0=0^x/0^x=0/0$, which is undefined

PS. I"ve read the explanation on magdalenarybarikova.comforum.org, but it isn"t clear khổng lồ me.

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In general, there is no good answer as khổng lồ what $0^0$ "should" be, so it is usually left undefined.

Basically, if you consider $x^y$ as a function of two variables, then there is no limit as $(x,y) o(0,0)$ (with $xgeq 0$): if you approach along the line $y=0$, then you get $limlimits_x o 0^+ x^0 = limlimits_x o 0^+ 1 = 1$; so perhaps we should define $0^0=1$? Well, the problem is that if you approach along the line $x=0$, then you get $limlimits_y o 0^+0^y = limlimits_y o 0^+ 0 = 0$. So should we define it $0^0=0$?

Well, if you approach along other curves, you"ll get other answers. Since $x^y = e^yln(x)$, if you approach along the curve $y=frac1ln(x)$, then you"ll get a limit of $e$; if you approach along the curve $y=fracln(7)ln(x)$, then you get a limit of $7$. & so on. There is just no good answer from the analytic point of view. So, for calculus & algebra, we just don"t want to give it any value, we just declare it undefined.

However, from a set-theory point of view, there actually is one & only one sensible answer to what $0^0$ should be! In phối theory, $A^B$ is the set of all functions from $B$ to lớn $A$; & when $A$ và $B$ denote "size" (cardinalities), then the "$A^B$" is defined to lớn be the kích thước of the mix of all functions from $A$ to lớn $B$. In this context, $0$ is the empty set, so $0^0$ is the collection of all functions from the empty set to the empty set. And, as it turns out, there is one (and only one) function from the empty set to the empty set: the empty function. So the set $0^0$ has one and only one element, và therefore we must define $0^0$ as $1$. So if we are talking about cardinal exponentiation, then the only possible definition is $0^0=1$, & we define it that way, period.

Added 2: the same holds in Discrete magdalenarybarikova.comematics, when we are mostly interested in "counting" things. In Discrete magdalenarybarikova.comematics, $n^m$ represents the number of ways in which you can make $m$ selections out of $n$ possibilities, when repetitions are allowed & the order matters. (This is really the same thing as "maps from $1,2,ldots,m$ khổng lồ $\1,2,ldots,n\$" when interpreted appropriately, so it is again the same thing as in mix theory).

So what should $0^0$ be? It should be the number of ways in which you can make no selections when you have no things lớn choose from. Well, there is exactly one way of doing that: just sit & do nothing! So we make $0^0$ equal khổng lồ $1$, because that is the correct number of ways in which we can bởi the thing that $0^0$ represents. (This, as opposed lớn $0^1$, say, where you are required lớn make $1$ choice with nothing to lớn choose from; in that case, you cannot do it, so the answer is that $0^1=0$).

Your "train of thoughts" don"t really work: If $x eq 0$, then $0^x$ means "the number of ways khổng lồ make $x$ choices from $0$ possibilities". This number is $0$. So for any number $k$, you have $kcdot 0^x = 0 = 0^x$, hence you cannot say that the equation $0^0cdot 0^x = 0^x$ suggests that $0^0$ "should" be $1$. The second argument also doesn"t work because you cannot divide by $0$, which is what you get with $0^x$ when $x eq 0$. So it really comes down to lớn what you want $a^b$ khổng lồ mean, and in discrete magdalenarybarikova.comematics, when $a$ & $b$ are nonnegative integers, it"s a count: it"s the number of distinct ways in which you can bởi a certain thing (described above), and that leads necessarily lớn the definition that makes $0^0$ equal to lớn $1$: because $1$ is the number of ways of making no selections from no choices.


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In the end, it is a matter of definition and utility. In Calculus và algebra, there is no reasonable definition (the closest you can come up with is trying lớn justify it via the binomial theorem or via nguồn series, which I personally think is a bit weak), and it is far more useful lớn leave it undefined or indeterminate, since otherwise it would lead to lớn all sorts of exceptions when dealing with the limit laws. In set theory, in discrete magdalenarybarikova.comematics, etc., the definition $0^0=1$ is both useful and natural, so we define it that way in that context. For other contexts (such as the one mentioned in magdalenarybarikova.comforum, when you are dealing exclusively with analytic functions where the problems with limits vị not arise) there may be both natural và useful definitions.

We basically define it (or fail to lớn define it) in whichever way it is most useful and natural to do so for the context in question. For Discrete magdalenarybarikova.comematics, there is no question what that "useful và natural" way should be, so we define it that way.