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by Quirkie on December 2nd, 2004

Quirkie

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What is 0 to the power of 0?

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Answers. 26 helpful answers below.

  • by Anonymous on September 14th, 2005

    Anonymous

    0^0 can be either 1 or 0. However, it is usually defined as 1.

    http://mathforum.org/dr.math/faq/faq.0.to.0.power.html

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  • by saratchandra on January 4th, 2005

    saratchandra

    Anything to the power of zero is 1 and zero to the power of anything is zero. As 0 to the power of zero takes both of these cases it is not mathematically defined

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  • by Farino on November 12th, 2007

    Farino

    0^0 = undefined
    Usually something to the power of 0 is 1 because going up in powers means multiplying by the base and going down in powers means dividing by the base.
    Going up:
    3^1 = 3
    3^2 = 3*3 = 9
    3^3 = 9*3 = 27 etc.
    Going down:
    3^3 = 27
    3^2 = 27/3 = 9
    3^1 = 9/3 = 3
    3^0 = 3/3 (base divided by itself) = 1

    Now let's apply that to 0:
    0^1 = 0
    0^0 = 0/0 = undefined (anything divided by 0 is undefined)

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  • by Peetee on November 12th, 2007

    Peetee

    Good question. The answer is undefined. In some cases zero is the better answer, in other cases one is the better answer. You need more information.

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  • by Woosel on August 30th, 2007

    Woosel

    In most cases it is reasonable to define it to be 1. In some contexts it may be 0 or undefined.

    The famous mathematician and computer scientist Donald Knuth explains in one of his books that 0^0 can be seen as special case

    - of x^0 (in which case it should be 1, because x^0 = 1 holds already for all other x)

    - of 0^x (in which case it should be 0, because 0^x=0 is true for all x>0)

    But he then points out that the function x^0 appears quite often in mathematics, for example in the Binomial Theorem, whereas the function 0^x is utterly unimportant. Therefore 0^0 should be seen as a special case of x^0, and hence it makes sense to set 0^0=1.

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  • by A-rod on April 29th, 2008

    A-rod

    It is mathematically undefined; anything to the power of 0 makes 1, 0 raised to anything it makes 0 thus, since these contradict each other mathematicians, have labeled it as undefined.

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  • by Magenta on November 12th, 2007

    Magenta

    Zero to any power is zero.
    Anything to the power of zero is zero.

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  • by Brian I on November 12th, 2007

    Brian I

    Zero

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  • by HDev is living On tHE EdgE on August 30th, 2007

    HDev is living On tHE EdgE

    0^0 can be written as 0^(x-x) or (0^x)/(0^x) . We know 0^x is 0 if x is not equal to zero. So, 0^0 comes out to be 0/0 which is indeterminate. It means that this value cannot be determined.

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  • by PerpetualAFK on March 7th, 2007

    PerpetualAFK

    0^0 is defined as 1.

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  • by Glenn Blaylock on December 2nd, 2004

    Glenn Blaylock

    0 to the power of 0 is the same as saying 0 divided by 0. So, see my answer to this question http://www.answerbag.com/c_view.php/427#q_11865

    In respnce to CheeseDude, it is too the same thing. Powers of 1 or more mean that you multiply a number by itself that many time. In other words, n^1 = n, n^2 = n * n, n^3 = n * n * n, etc. Powers that are less than 1 mean that you divide such that n^0 = n/n, n^-1 = n/n/n, etc. Therefore, 0^0 = 0/0. Thus my answer.

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  • by Peetee on May 8th, 2009

    Peetee

    Undefined. You can't tell unless you know that it is part of a greater function (eg. x^0 or 0^x). Then you have to look at how the function works in the real world. Is it continuous? (or should it be continuous?) Then use the answer that makes it continuous. If not, then it is still undefined. Don't rule out having two answers ( eg 0 and 1). The short answer is that 0^0 is undefined. You need more information.

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  • by 92kenneth on January 31st, 2009

    92kenneth

    i am confused.i tried to calculate 0^0 but my calculator says "math error" which means no solution.so i guess it should be infinity.

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  • by ameina on January 17th, 2009

    ameina

    Infinite.

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  • by john on April 17th, 2008

    john

    1

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  • by bloodlylilcorpse is back on May 8th, 2009

    bloodlylilcorpse is back

    I think its a cute lil' anime smiley face! ^_^

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  • by H on August 30th, 2007

    H

    I do not know for certain, but as a general rule n^0 = 1.

    I would strongly argue that 0^0 is 1, however.


    You could test it graphically, I guess, y=x^0 y=1 for all values of x... hmmmmmm

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  • by Obey Gravity on November 14th, 2007

    Obey Gravity

    NO. ALL OF THESE ARE WRONG. Any number to the 0 power is NEVER zero. Zero to the zero power is NOT undefined. Any number OTHER than zero "powered" by zero is undefined. But, If ZERO is powered to ZERO it is INDETERMINATE, INDETERMINATE, INDETERMINATE!!

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  • by DA BEN DAN yanggui zi on October 5th, 2011

    DA BEN DAN yanggui zi

    the answer is undefined.

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  • by Fakhr_R on October 5th, 2011

    Fakhr_R

    ask my math teacher

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  • by Anonymous on December 4th, 2008

    Anonymous

    Zero,my classmate Dana Ikeda would say,"0/0=0",she also said. It may be one, but 0/0 could =1.0x0=0 0/0=?

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  • by Advolo on April 3rd, 2009

    Advolo

    Conclusion : [ 0^0 ] = 0 or 1

    If, zero = number
    [^^^] = Definition of Liuhui Brahmagupta
    = [ ( 1 - 1 ) = ( 0 ) ]
    = [ ( Living Zero ) = ( Dead Zero ) ]
    = [ (0_) = (_0) ]

    [ 0 ÷ 0 ] = [ (_0) ÷ (_0) ] = [ (_0) ] = [ 0 ]
    [ 0 ÷ 0 ] = [ (0_) ÷ (_0) ] = [ (_0) ] = [ 0 ]
    [ 0 ÷ 0 ] = [ (_0) ÷ (0_) ] = [ (_0) ] = [ 0 ]
    [ 0 ÷ 0 ] = [ (0_) ÷ (0_) ] = [ 1 ]

    [ 0 ÷ 0 ] = [ (_0) ÷ (_0) ] = [ (_0) ] = [ 0 ]
    [ 0 ÷ 0 ] = [ ( 1 - 1 ) ÷ (_0) ] = [ (_0) ] = [ 0 ]
    [ 0 ÷ 0 ] = [ (_0) ÷ ( 1 - 1 ) ] = [ (_0) ] = [ 0 ]
    [ 0 ÷ 0 ] = [ ( 1 - 1 ) ÷ ( 1 - 1 ) ] = [ 1 ]

    [ 0^0 ] = [ (_0)^(_0) ] = [ (_0) ] = [ 0 ]
    [ 0^0 ] = [ (_0)^(0_) ] = [ (_0) ] = [ 0 ]
    [ 0^0 ] = [ (0_)^(_0) ] = [ (_0) ] = [ 0 ]
    [ 0^0 ] = [ (0_)^(0_) ] = [ 1 ]

    [ 0^0 ] = [ (_0)^(_0) ] = [ (_0) ] = [ 0 ]
    [ 0^0 ] = [ (_0)^( 1 - 1 ) ] = [ (_0) ] = [ 0 ]
    [ 0^0 ] = [ ( 1 - 1 )^(_0) ] = [ (_0) ] = [ 0 ]
    [ 0^0 ] = [ ( 1 - 1 )^( 1 - 1 ) ] = [ 1 ]

    [ 0^0 ] = [ 0 ÷ 0 ]^[ 0 ÷ 0 ] = (_0)^(_0) = (_0) = 0
    [ 0^0 ] = [ 0 ÷ 0 ]^[ 0 ÷ 0 ] = (_0)^1 = (_0) = 0
    [ 0^0 ] = [ 0 ÷ 0 ]^[ 0 ÷ 0 ] = 1^(_0) = (_0) = 0
    [ 0^0 ] = [ 0 ÷ 0 ]^[ 0 ÷ 0 ] = 1^1 = 1

    Coupdetat.net (2009.04.04)

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  • by Dontbeignorant on May 8th, 2009

    Dontbeignorant

    if 0 to the power of 0 is 0 then would this rule apply?
    (x^x)(x^y)=x^(x+y)
    x^(x+y) must be greater than or equal to x^x
    x^(x+y) must be greater than or equal to x^y
    works with every number but 0, when zero=one.
    (0^0)(0^1)=0^(0+1)=0^1=0, yet 0^0=1?
    (1^1)(1^0)=1^(1+0)=1^1=1, 1^0=1, 1^1=1
    (2^2)(2^0)=2^(2+0)=2^2=4, 2^0=1, 2^2=4
    (2^2)(2^1)=2^(2+1)=2^3=8, 2^2=4, 2^1=2

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  • by Haoie on May 8th, 2009

    Haoie

    The answer is 1.

    Anything to the power of 0 is 1.

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  • by Mr_Natural Abstractor of the Quintessence on May 8th, 2009

    Mr_Natural Abstractor of the Quintessence

    0^0 is undefined, just as n/0 is undefined. That's the "most likely" answer.
    ====================================
    Addendum:
    "The debate has been going on at least since the early 1800s. At that time, most mathematicians agreed that 0^0 = 1, but in 1821 Cauchy listed 0^0 along with expressions like 0/0 in a table of undefined forms."
    ~Wikipedia discussion:
    http://en.wikipedia.org/wiki/0%5E0#Zero_to_the_zero_power

    On the same page, skip down to "Treatment in programming languages and calculators":
    http://en.wikipedia.org/wiki/0%5E0#Treatment_in_programming_languages_and_calculators

    (Various calculators and programming languages treat 0^0 differently.)

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  • by Mohammad_hadi94 on November 11th, 2010

    Mohammad_hadi94

    0*111111111111111111111111111111111111111111111111111111111=0
    0*999999999999999999999999999999999999999999999999999999999999*(99999^9999)=0
    so what ever the number is huge when multiply by 0 it will be equal to zero
    then 0*(infinity)=0
    0/0=??
    multiply numerator and demenator by 1
    0/0=(0*1)/(0*1)=(0/1)*(1/0)=0*(infinity)=0
    hence 0/0=0

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