0^0 "the 0th power of 0" has no value, I guess. How do you deal with it?

Answer 1

Sometimes, it is convenient to define 0^0 = 1, but it could break some mathematical rules if not careful.

We generally regard it as being indeterminate just like 0/0, infty - infty, and infty/infty.

Some other indeterminate forms are 1^infty and infty^0.

In calculus, for instance, consider the limit:

lim[x->L] f(x)^g(x), where lim[x->L] f(x) = lim[x->L] g(x) = 0, this is basically a limit of the form 0^0. Most of the time, when you encounter this type of limit, you will find the limit to be 1, BUT that is not always the case.

If I find a counter-example before this question closes, I will post it.

[Er, um except for the above poster's example of 0^x, which I just now saw]

Ah yes, take f(x)=e^(-1/x^2) and g(x)=x.

(Interesting fact for you calc buffs, if you define f(0)=0 with my above example, f will become continuous and infinitely differentiable at 0. Not only that, but every derivative at 0 is 0. This means the Maclaurin Series this monster is a big fat zero.)

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Edit: New thought, about 4 hours later...

Even when we examine limits of the 0^0 type, I don't think defining 0^0 to be 1 would be inconsistent because it would be a *discontinuity* of a function f(x,y) = x^y at the point (0,0), and we Calc students should know that at removable discontinuities where the function is defined, the limit *isn't* the value of the function. Go ahead and try to picture the graph of z = x^y, defined for any x and y where either x is positive, or x=0 and y>=0. The graph is actually a continuous surface, with the exception being at (0,0), which would be one of those kinks. And remember, when dealing with a function of two variables, we can approach a point from infinitely many directions, not from a positive or negative side.

Thus, I think we can still define 0^0=1 and not violate limit laws.

Answer 2

I must concur with Sam and Jim P above -- 0^0=1. This is the algebraic definition, which occurs prior to any kind of extension of the exponential function by continuity or the Taylor series for e^x. Supporting arguments for this definition:

http://www.faqs.org/faqs/sci-math-faq/specialnumbers/0to0/
http://en.wikipedia.org/wiki/Exponentiation#Zero_to_the_zero_power

The arguments for this definition are numerous, whereas the arguments against defining it come in exactly two forms:

#1: a misconception that the value of x^0 was arrived at only by formally applying the laws of exponents -- to wit, a^0 = a^(1-1) = a^1/a^1 = a/a = 1, which obviously only works when a≠0. However, the rule a^0 = 1 can be justified far more naturally by regarding it as the product of no numbers, or as the number of functions from the empty set to a set with a elements, both of which require 0^0=1.

#2: Noting that [x→0⁺]lim 0^x = 0 and [x→0]lim x^0 = 1 ≠ 0, so that no value of 0^0 can be assigned that would make the exponential function continuous there. However, there is no essential reason why we should require the exponential function to be continuous. It certainly doesn't help matters -- limits of the form 0^0 will require special analysis whether we actually define 0^0 or not, so the only thing that this does is force us to lengthen the identities for the binomial theorem and the exponential function (i.e. the usual definition of e^x = [k=0, ∞]∑x^k/k! doesn't work unless 0^0 is defined to be 1).

In general, I would argue that any mathematical rules that might require exceptions if we defined 0^0 = 1 _already_ require exceptions (i.e. the rule about 0^(n-m) = 0^n/0^m already doesn't work - 0 = 0^1 = 0^(2-1) ≠ 0^2/0^1 = 0/0, limits of the form [x→c]lim f(x)^g(x) where [x→c]lim f(x) = [x→c]lim g(x) = 0 already require special treatment, defining 0^0=1 won't change that), whereas NOT defining 0^0=1 breaks many formulas that are already valid in general (the aforementioned binomial theorem and taylor series, as well as several other combinatoric identities, such as a^b being the number of b-tuples of elements from a set of a elements).

Further, it deprives teachers of the infinitely useful pedagogical tool of a commonly used, _natural_ function (i.e. not something defined piecewise) that contains an essential discontinuity (which helps to refute the notion that finding limits means to cancel stuff until the function is defined at the limiting value and then substituting), as well as a chance to explain the difference between a value that is undefined and one that is indeterminate. It's important to explain that difference, because these are - gasp - not the same thing! Undefined is a property of the domain of the function (namely, that it doesn't include that point), whereas indeterminate is a property of the limits of that function as you approach that point (namely, that they don't exist). And if students don't understand that indeterminate doesn't imply undefined, or vice versa, they miss out on valuable understanding. I have heard many people repeat the untruth that, say, "0/0 is not undefined, it's actually indeterminate." Of course, half of that statement is true -- 0/0 really is indeterminate, since no information can be derived about the limit of [x→c]lim f(x)/g(x) from the limits of f(x) and g(x) if [x→c]lim f(x) = 0 and [x→c]lim g(x) = 0. But it is also undefined, since there is no algebraic interpretation of 0/0 which would be consistent with the field axioms. Conversely, there IS an algebraic interpretation of 0^0 which is consistent with the additive, multiplicative, and combinatorial properties of exponentiation, and that is 0^0=1.

Now, of course, in mathematics, you are free to define (or to fail to define) any expression in any manner you choose, within the bounds of consistency. So it's impossible to "prove" that 0^0 must be defined as 1, any more than it's possible to prove that a+b is the sum of a and b. But it makes very little sense to leave it undefined, and I feel that the mathematics textbooks that do so, only do so _because_ they don't want to have to explain mathematics to students. After all, if we told them that the definition of many mathematical functions is decided based on convenience rather than logical necessity, and that these definitions can differ from author to author, they might begin actually thinking about the material they're learning instead of just waiting for the teacher to spoon-feed it to them, and we can't have that, can we?

Answer 3

x to the power 0 is 1, see 1 to the power 0 =1 it implies 0th root of 1 = 1 or 0 to the power 1/0 =1 and 1/0 = infinity and 1 to the power infinity =1. Now let us see this in case when x=0 it means 0 to the power 0 =1 or 0th root of 0 = 1 but 0th root of 1 = 1 it implies 0 to the power 0 is not equal to 1 but their exists a number 1/infinity which is equal to '0 to the power 0'.

Answer 4

log 0=a million because of the fact as an occasion log 0(base 10) =x 10(capability)x=0 10(capability)a million=0 subsequently log 0=a million enable log a million(base 10)=x 10(capability)x=a million we can't get the x capability to get a million subsequently log a million=undefined

Answer 5

Although I think some people claim that is generally given a value (0? 1?) by convention, I would personally agree that it has no value (is undefined).

On one hand, lim x^0 as x->0 is clearly 1.
Also, lim x^x as x->0+ is also 1.
But lim 0^x as x->0 is clearly 0.

Answer 6

If you examine lim x-->0 of x^x from the right, you get a value of 1.

However, if you examine that limit from the left you will find that it isn't defined.

Therefore, since the left- and right-hand limits do not match 0^0 is not defined.

Answer 7

Usually, anything to the zero power is 1, however, 0^0 doesn't exist...

0^0 doesn't equal 1.

You should never see this, except as a trick question to which you would answer, doesn't exist...

Answer 8

0^0 = (0^1)/(0^1)
= 0/0

0/0 is indeterminate as it can have any value.

Answer 9

Sam is right. Any number to the "0"th power is "1". (Search Yahoo and Google for "0^0".)

I don't understand how someone with the right answer can get so many thumbs down. Does anyone think before they vote?

Answer 10

0^0 is equal to 1, anything to the zero power is equal to 1.