What is 0 to the 0 power?

Answer 1

According to some Calculus textbooks, 0^0 is an "indeterminate form." What mathematicians mean by "indeterminate form" is that in some cases we think about it as having one value, and in other cases we think about it as having another.

When evaluating a limit of the form 0^0, you need to know that limits of that form are "indeterminate forms," and that you need to use a special technique such as L'Hopital's rule to evaluate them. For instance, when evaluating the limit Sin[x]^x (which is 1 as x goes to 0), we say it is equal to x^x (since Sin[x] and x go to 0 at the same rate, i.e. limit as x->0 of Sin[x]/x is 1). Then we can see from the graph of x^x that its limit is 1.

Other than the times when we want it to be indeterminate, 0^0 = 1 seems to be the most useful choice for 0^0 . This convention allows us to extend definitions in different areas of mathematics that would otherwise require treating 0 as a special case. Notice that 0^0 is a discontinuity of the function f(x,y) = x^y, because no matter what number you assign to 0^0, you can't make x^y continuous at (0,0), since the limit along the line x=0 is 0, and the limit along the line y=0 is 1.

This means that depending on the context where 0^0 occurs, you might wish to substitute it with 1, indeterminate or undefined/nonexistent.

Some people feel that giving a value to a function with an essential discontinuity at a point, such as x^y at (0,0), is an inelegant patch and should not be done. Others point out correctly that in mathematics, usefulness and consistency are very important, and that under these parameters 0^0 = 1 is the natural choice.

The following is a list of reasons why 0^0 should be 1.

Rotando & Korn show that if f and g are real functions that vanish at the origin and are analytic at 0 (infinitely differentiable is not sufficient), then f(x)^g(x) approaches 1 as x approaches 0 from the right.

From Concrete Mathematics p.162 (R. Graham, D. Knuth, O. Patashnik):

Some textbooks leave the quantity 0^0 undefined, because the functions 0^x and x^0 have different limiting values when x decreases to 0. But this is a mistake. We must define x^0=1 for all x , if the binomial theorem is to be valid when x=0 , y=0 , and/or x=-y . The theorem is too important to be arbitrarily restricted! By contrast, the function 0^x is quite unimportant.

Published by Addison-Wesley, 2nd printing Dec, 1988.

As a rule of thumb, one can say that 0^0 = 1 , but 0.0^(0.0) is undefined, meaning that when approaching from a different direction there is no clearly predetermined value to assign to 0.0^(0.0) ; but Kahan has argued that 0.0^(0.0) should be 1, because if f(x), g(x) --> 0 as x approaches some limit, and f(x) and g(x) are analytic functions, then f(x)^g(x) --> 1 .

The discussion of 0^0 is very old. Euler argues for 0^0 = 1 since a^0 = 1 for a not equal to 0 . The controversy raged throughout the nineteenth century, but was mainly conducted in the pages of the lesser journals: Grunert's Archiv and Schlomilch's Zeitshrift. Consensus has recently been built around setting the value of 0^0 = 1..

Answer 2

1

Answer 3

1

Answer 4

Any number with the power 0 the answer is 1

Answer 5

well, its undefined. take a look at this.

0 to the power of any number gives the ans as 0.

any number to the power of 0 gives you the answer as 1.

so, what is 0^0?? undefined.

Answer 6

As per laws of indices a^0 = 1
here a =0 therefore 0^0 = 1

Answer 7

No...naah..not undefined!!

See, anything to the power of 0 is 1.
But 0 itself is smaller than 1 whole number. Therefore 0^0 = 0. As it is smaller than 1 and if it is to the power of 0, it means nil so it is 0.

Forget what calculator says... Calculator says 10000000000000 * 100000000000 = 1.33333E so will it be infinity, no it will be 10 to the power of 24.

Answer 8

Its undefined. It is indeterminate, i.e. it cannot be determined.

For every function which reduces to 0 to the power 0, you can use L'Hospital's rule.

Some more events where the value is considered undefined are:::

0/0, infinite/infinite, infinite - infinite, 0 x infinite, anything to the power infinite, 0 to the power 0, infinite to the power 0.

Answer 9

usually with numbers it should be 1 but for 0 its o

Answer 10

The answer is -- it depends how it is used.

Look at the wikipedia page here

http://en.wikipedia.org/wiki/0%5E0#Zero_to_the_zero_power

In many settings, 00 is defined to be 1. This definition arises in foundational treatments of the natural numbers as finite cardinals, and is useful for shortening combinatorial identities and removing special cases from theorems, as illustrated below. In many other settings, 00 is left undefined. In calculus, 00 is an indeterminate form, which must be analyzed rather than evaluated. In general, mathematical analysis treats 00 as undefined[3] in order that the exponential function be continuous

Justifications for defining 00 = 1 include:

When 00 is regarded as an empty product of zeros, its value is 1.
The combinatorial interpretation of 00 is the number of empty tuples of elements from the empty set. There is exactly one empty tuple.
Equivalently, the set-theoretic interpretation of 00 is the number of functions from the empty set to the empty set. There is exactly one such function, the empty function.
A power series identity with nonzero constant term

and it goes on and on read the article