zero to the power of zero

Many widely used formulas involving natural-number exponents require {{math|0⁰}} to be defined as {{math|1}}. For example, the following three interpretations of {{math|b{{sup|0}}}} make just as much sense for {{math|1=b = 0}} as they do for positive integers {{mvar|b}}:

• The interpretation of {{math|b{{sup|0}}}} as an empty product assigns it the value {{math|1}}.
• The combinatorial interpretation of {{math|b⁰}} is the number of 0-tuples of elements from a {{math|b}}-element set; there is exactly one 0-tuple.
• The set-theoretic interpretation of {{math|b⁰}} is the number of functions from the empty set to a {{math|b}}-element set; there is exactly one such function, namely, the empty function.

All three of these specialize to give {{math|1=0{{sup|0}} = 1}}.

When evaluating polynomials, it is convenient to define {{math|0⁰}} as {{math|1}}. A (real) polynomial is an expression of the form {{math|a₀x⁰ + ⋅⋅⋅ + a_nxⁿ}}, where {{math|x}} is an indeterminate, and the coefficients {{math|a_i}} are real numbers. Polynomials are added termwise, and multiplied by applying the distributive law and the usual rules for exponents. With these operations, polynomials form a ring {{math|R[x]}}. The multiplicative identity of {{math|R[x]}} is the polynomial {{math|x⁰}}; that is, {{math|x⁰}} times any polynomial {{math|p(x)}} is just {{math|p(x)}}. Also, polynomials can be evaluated by specializing {{math|x}} to a real number. More precisely, for any given real number {{math|r}}, there is a unique unital algebra homomorphism {{math|ev_r : R[x] → R}} such that {{math|1=ev_r(x) = r}}. Because {{math|ev_r}} is unital, {{math|1=ev_r(x⁰) = 1}}. That is, {{math|1=r⁰ = 1}} for each real number {{math|r}}, including 0. The same argument applies with {{math|R}} replaced by any ring.

Defining {{math|1=0⁰ = 1}} is necessary for many polynomial identities. For example, the binomial theorem $(1+x)^\{n\}=\backslash sum\_\{k=0\}^\{n\}\backslash binom\{n\}\{k\}x^\{k\}$ holds for {{math|1=x = 0}} only if {{math|1=0⁰ = 1}}.

Similarly, rings of power series require {{math|1=x⁰}} to be defined as 1 for all specializations of {{math|x}}. For example, identities like $\backslash frac\{1\}\{1-x\}=\backslash sum\_\{n=0\}^\{\backslash infty\}x^n$ and $e^x=\backslash sum\_\{n=0\}^\{\backslash infty\}\backslash frac\{x^n\}\{n!\}$ hold for {{math|1=x = 0}} only if {{math|1=0⁰ = 1}}.

In order for the polynomial {{math|x⁰}} to define a continuous function {{math|R → R}}, one must define {{math|1=0⁰ = 1}}.

In calculus, the power rule $\backslash frac\{d\}\{dx\}x^n=nx^\{n-1\}$ is valid for {{math|1=n = 1}} at {{math|1=x = 0}} only if {{math|1=0⁰ = 1}}.

Image:X^y.png

Limits involving algebraic operations can often be evaluated by replacing subexpressions with their limits; if the resulting expression does not determine the original limit, the expression is known as an indeterminate form. The expression {{math|0{{sup|0}}}} is an indeterminate form: Given real-valued functions {{math|f(t)}} and {{math|g(t)}} approaching {{math|0}} (as {{math|t}} approaches a real number or {{math|±∞}}) with {{math|f(t) > 0}}, the limit of {{math|f(t)^g(t)}} can be any non-negative real number or {{math|+∞}}, or it can diverge, depending on {{mvar|f}} and {{mvar|g}}. For example, each limit below involves a function {{math|f(t)^g(t)}} with {{math|f(t), g(t) → 0}} as {{math|t → 0⁺}} (a one-sided limit), but their values are different:

$$\backslash lim\_\{t\; \backslash to\; 0^+\}\; \{t\}^\{t\}\; =\; 1\; ,$$

$$\backslash lim\_\{t\; \backslash to\; 0^+\}\; \backslash left(e^\{-1/t^2\}\backslash right)^t\; =\; 0,$$

$$\backslash lim\_\{t\; \backslash to\; 0^+\}\; \backslash left(e^\{-1/t^2\}\backslash right)^\{-t\}\; =\; +\backslash infty,$$

$$\backslash lim\_\{t\; \backslash to\; 0^+\}\; \backslash left(a^\{-1/t\}\backslash right)^\{-t\}\; =\; a.$$

Thus, the two-variable function {{math|x{{i sup|y}}}}, though continuous on the set {{math|{(x, y) : x > 0}{{null}}}}, cannot be extended to a continuous function on {{math|{(x, y) : x > 0} ∪ {(0, 0)}{{null}}}}, no matter how one chooses to define {{math|0⁰}}.

On the other hand, if {{math|f}} and {{math|g}} are analytic functions on an open neighborhood of a number {{mvar|c}}, then {{math|f(t)^g(t) → 1}} as {{mvar|t}} approaches {{mvar|c}} from any side on which {{mvar|f}} is positive.

This and more general results can be obtained by studying the limiting behavior of the function $\backslash log(f(t)^\{g(t)\})=g(t)\backslash log\; f(t)$.{{cite journal

| last1 = Baxley

| first1 = John V.

| last2 = Hayashi

| first2 = Elmer K.

| date = June 1978

| title = Indeterminate Forms of Exponential Type

| url = https://www.jstor.org/stable/2320074

| journal = The American Mathematical Monthly

| volume = 85

| issue = 6

| pages = 484–486

| doi = 10.2307/2320074

| jstor = 2320074

| access-date = 23 November 2021

}}{{cite journal

| last1 = Xiao

| first1 = Jinsen

| last2 = He

| first2 = Jianxun

| date = December 2017

| title = On Indeterminate Forms of Exponential Type

| url = https://www.jstor.org/stable/10.4169/math.mag.90.5.371

| journal = Mathematics Magazine

| volume = 90

| issue = 5

| pages = 371–374

| doi = 10.4169/math.mag.90.5.371

| jstor = 10.4169/math.mag.90.5.371

| s2cid = 125602000

| access-date = 23 November 2021

}}

In the complex domain, the function {{math|z{{i sup|w}}}} may be defined for nonzero {{math|z}} by choosing a branch of {{math|log z}} and defining {{math|z{{i sup|w}}}} as {{math|e{{i sup|w log z}}}}. This does not define {{math|0^w}} since there is no branch of {{math|log z}} defined at {{math|1=z = 0}}, let alone in a neighborhood of {{math|0}}.

In 1752, Euler in Introductio in analysin infinitorum wrote that {{math|1=a{{sup|0}} = 1}} and explicitly mentioned that {{math|1=0{{sup|0}} = 1}}. An annotation attributed to Mascheroni in a 1787 edition of Euler's book Institutiones calculi differentialis offered the "justification"

$$0^0\; =\; (a-a)^\{n-n\}\; =\; \backslash frac\{(a-a)^n\}\{(a-a)^n\}\; =\; 1$$

as well as another more involved justification. In the 1830s, Libri published several further arguments attempting to justify the claim {{math|1=0⁰ = 1}}, though these were far from convincing, even by standards of rigor at the time.

Euler, when setting {{math|1=0{{sup|0}} = 1}}, mentioned that consequently the values of the function {{math|0{{sup|x}}}} take a "huge jump", from {{math|∞}} for {{math|x < 0}}, to {{math|1}} at {{math|1=x = 0}}, to {{math|0}} for {{math|x > 0}}.

In 1814, Pfaff used a squeeze theorem argument to prove that {{math|x{{sup|x}} → 1}} as {{math|x → 0{{sup|+}}}}.

On the other hand, in 1821 Cauchy explained why the limit of {{math|x{{sup|y}}}} as positive numbers {{mvar|x}} and {{mvar|y}} approach {{math|0}} while being constrained by some fixed relation could be made to assume any value between {{math|0}} and {{math|∞}} by choosing the relation appropriately. He deduced that the limit of the full two-variable function {{math|x{{sup|y}}}} without a specified constraint is "indeterminate". With this justification, he listed {{math|0⁰}} along with expressions like {{math|{{sfrac|0|0}}}} in a table of indeterminate forms.

Apparently unaware of Cauchy's work, Möbius in 1834, building on Pfaff's argument, claimed incorrectly that {{math|f(x)^g(x) → 1}} whenever {{math|f(x),g(x) → 0}} as {{mvar|x}} approaches a number {{mvar|c}} (presumably {{mvar|f}} is assumed positive away from {{mvar|c}}). Möbius reduced to the case {{math|1=c = 0}}, but then made the mistake of assuming that each of {{mvar|f}} and {{mvar|g}} could be expressed in the form {{math|Px{{sup|n}}}} for some continuous function {{mvar|P}} not vanishing at {{math|0}} and some nonnegative integer {{mvar|n}}, which is true for analytic functions, but not in general. An anonymous commentator pointed out the unjustified step; then another commentator who signed his name simply as "S" provided the explicit counterexamples {{math|(e^−1/x)^x → e{{sup|−1}}}} and {{math|(e^−1/x)^2x → e{{sup|−2}}}} as {{math|x → 0{{sup|+}}}} and expressed the situation by writing that "{{math|0{{sup|0}}}} can have many different values".

• Some authors define {{math|0⁰}} as {{math|1}} because it simplifies many theorem statements. According to Benson (1999), "The choice whether to define {{math|0⁰}} is based on convenience, not on correctness. If we refrain from defining {{math|0⁰}}, then certain assertions become unnecessarily awkward. ... The consensus is to use the definition {{math|1=0⁰ = 1}}, although there are textbooks that refrain from defining {{nowrap begin}}{{math|0⁰}}."{{nowrap end}} Knuth (1992) contends more strongly that {{math|0⁰}} "has to be {{math|1}}"; he draws a distinction between the value {{math|0⁰}}, which should equal {{math|1}}, and the limiting form {{math|0⁰}} (an abbreviation for a limit of {{math|f(t)^g(t)}} where {{math|f(t), g(t) → 0}}), which is an indeterminate form: "Both Cauchy and Libri were right, but Libri and his defenders did not understand why truth was on their side."
• Other authors leave {{math|0⁰}} undefined because {{math|0⁰}} is an indeterminate form: {{math|f(t), g(t) → 0}} does not imply {{math|f(t)^g(t) → 1}}.

There do not seem to be any authors assigning {{math|0⁰}} a specific value other than 1.

The IEEE 754-2008 floating-point standard is used in the design of most floating-point libraries. It recommends a number of operations for computing a power:

• pown (whose exponent is an integer) treats {{math|0⁰}} as {{math|1}}; see {{section link||Discrete exponents}}.
• pow (whose intent is to return a non-NaN result when the exponent is an integer, like pown) treats {{math|0⁰}} as {{math|1}}.
• powr treats {{math|0⁰}} as NaN (Not-a-Number) due to the indeterminate form; see {{section link||Continuous exponents}}.

The pow variant is inspired by the pow function from C99, mainly for compatibility. It is useful mostly for languages with a single power function. The pown and powr variants have been introduced due to conflicting usage of the power functions and the different points of view (as stated above).

The C and C++ standards do not specify the result of {{math|0⁰}} (a domain error may occur). But for C, as of C99, if the normative annex F is supported, the result for real floating-point types is required to be {{math|1}} because there are significant applications for which this value is more useful than NaN (for instance, with discrete exponents); the result on complex types is not specified, even if the informative annex G is supported. The Java standard, the .NET Framework method System.Math.Pow, Julia, and Python also treat {{math|0⁰}} as {{math|1}}. Some languages document that their exponentiation operation corresponds to the pow function from the C mathematical library; this is the case with Lua's ^ operator and Perl's ** operator (where it is explicitly mentioned that the result of 0**0 is platform-dependent).

R, SageMath, and PARI/GP evaluate {{math|{{var|x}}{{sup|0}}}} to {{math|1}}. Mathematica simplifies {{math|{{var|x}}{{sup|0}}}} to {{math|1}} even if no constraints are placed on {{math|{{var|x}}}}; however, if {{math|0{{sup|0}}}} is entered directly, it is treated as an error or indeterminate. Mathematica and PARI/GP further distinguish between integer and floating-point values: If the exponent is a zero of integer type, they return a {{math|1}} of the type of the base; exponentiation with a floating-point exponent of value zero is treated as undefined, indeterminate or error.

• 0/0

{{reflist|refs=

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{{cite book |author-first=Nicolas |author-last=Bourbaki |title=Algèbre |publisher=Springer |date=1970 |author-link=Nicolas Bourbaki |chapter=§III.2 No. 9 |quote=L'unique monôme de degré {{math|0}} est l'élément unité de {{math|A[(X_i)_i∈I]}}; on l'identifie souvent à l'élément unité {{math|1}} de {{math|A}}}}

{{cite book |author-first=Nicolas |author-last=Bourbaki |title=Algèbre |publisher=Springer |date=1970 |author-link=Nicolas Bourbaki |chapter=§IV.1 No. 3}}

{{cite journal |author-first=A. F. |author-last=Möbius |title=Beweis der Gleichung {{math|1=0⁰ = 1}}, nach J. F. Pfaff |trans-title=Proof of the equation {{math|1=0⁰ = 1}}, according to J. F. Pfaff |language=de |url=http://babel.hathitrust.org/cgi/pt?id=mdp.39015036988163;view=1up;seq=142 |journal=Journal für die reine und angewandte Mathematik |issue=12 |date=1834 |pages=134–136 |doi=10.1515/crll.1834.12.134 |volume=1834 |s2cid=199547186}}

{{cite book |date=1988 |title=Introduction to analysis of the infinite, Book 1 |author-first=Leonhard |author-last=Euler |author-link=Leonhard Euler |translator-first=J. D. |translator-last=Blanton |publisher=Springer |isbn=978-0-387-96824-7 |url=https://books.google.com/books?id=H58dmcLEnk4C |chapter=Chapter 6, §97 |page=75}}

{{cite journal |author-last=Libri |author-first=Guillaume |author-link=Guglielmo Libri Carucci dalla Sommaja |title=Mémoire sur les fonctions discontinues. |language=fr |journal=Journal für die reine und angewandte Mathematik |date=1833 |volume=1833 |issue=10 |pages=303–316 |doi=10.1515/crll.1833.10.303 |s2cid=121610886 |url=http://www.numdam.org/item/ASENS_1875_2_4__57_0/}}

{{cite journal |author-last=Knuth |author-first=Donald E. |author-link=Donald E. Knuth |title=Two Notes on Notation |journal=The American Mathematical Monthly |date=1992 |volume=99 |issue=5 |pages=403–422 |doi=10.1080/00029890.1992.11995869 |arxiv=math/9205211 |bibcode=1992math......5211K}}

{{cite journal |author=Anonymous |title=Bemerkungen zu dem Aufsatze überschrieben "Beweis der Gleichung {{math|1=0⁰ = 1}}, nach J. F. Pfaff" |trans-title=Remarks on the essay "Proof of the equation {{math|1=0⁰ = 1}}, according to J. F. Pfaff" |language=de |url=http://babel.hathitrust.org/cgi/pt?id=mdp.39015036988163;view=1up;seq=300 |journal=Journal für die reine und angewandte Mathematik |issue=12 |date=1834 |pages=292–294 |doi=10.1515/crll.1834.12.292 |volume=1834}}

{{cite book |author-first=Donald C. |author-last=Benson |title=The Moment of Proof: Mathematical Epiphanies |location=New York, USA |publisher=Oxford University Press |publication-place=Oxford, UK |date=1999 |page=29 |isbn=978-0-19-511721-9}}

{{cite book |author-last1=Muller |author-first1=Jean-Michel |author-last2=Brisebarre |author-first2=Nicolas |author-last3=de Dinechin |author-first3=Florent |author-last4=Jeannerod |author-first4=Claude-Pierre |author-last5=Lefèvre |author-first5=Vincent |author-last6=Melquiond |author-first6=Guillaume |author-last7=Revol |author-first7=Nathalie |author-link7=Nathalie Revol |author-last8=Stehlé |author-first8=Damien |author-last9=Torres |author-first9=Serge |title=Handbook of Floating-Point Arithmetic |date=2010 |publisher=Birkhäuser |edition=1 |doi=10.1007/978-0-8176-4705-6 |lccn=2009939668|page=216 |isbn=978-0-8176-4704-9 |s2cid=5693480 |url=https://cds.cern.ch/record/1315760}} {{isbn|978-0-8176-4705-6}} (online), {{isbn|0-8176-4704-X}} (print)

{{cite web |url=https://reference.wolfram.com/language/ref/Power.html |title=Wolfram Language & System Documentation: Power |publisher=Wolfram | access-date=2018-08-02}}

{{cite web |url=http://pari.math.u-bordeaux.fr/cgi-bin/gitweb.cgi?p=pari.git;a=commitdiff;h=c2fac9a15 |title=pari.git / commitdiff – 10- x ^ t_FRAC: return an exact result if possible; e.g. 4^(1/2) is now 2 |access-date=2018-09-10}}

{{cite book |title=Concrete Mathematics |edition=1st |publisher=Addison-Wesley Longman Publishing Co. |date=1989-01-05 |isbn=0-201-14236-8 |author-first=Ronald |author-last=Graham |author-link=Ronald Graham |author-first2=Donald |author-last2=Knuth |author-link2=Donald Knuth |author-first3=Oren |author-last3=Patashnik |author-link3=Oren Patashnik |page=162 |chapter=Binomial coefficients |title-link=Concrete Mathematics |quote=Some textbooks leave the quantity {{math|0⁰}} undefined, because the functions {{math|x⁰}} and {{math|0^x}} have different limiting values when {{math|x}} decreases to 0. But this is a mistake. We must define {{math|1=x⁰ = 1}}, for all {{math|x}}, if the binomial theorem is to be valid when {{math|1=x = 0}}, {{math|1=y = 0}}, and/or {{math|1=x = −y}}. The binomial theorem is too important to be arbitrarily restricted! By contrast, the function {{math|0^x}} is quite unimportant.}}

{{cite journal |author-last1=Vaughn |author-first1=Herbert E. |date=1970 |title=The expression {{math|0{{sup|0}}}} |journal=The Mathematics Teacher |volume=63 |issue= |pages=111–112}}

{{cite book |author-first1=S. C. |author-last1=Malik |author-first2=Savita |author-last2=Arora |date=1992 |title=Mathematical Analysis |publisher=Wiley |location=New York, USA |page=223 |isbn=978-81-224-0323-7 |quote=In general the limit of {{math|φ(x)/ψ(x)}} when {{math|1=x = a}} in case the limits of both the functions exist is equal to the limit of the numerator divided by the denominator. But what happens when both limits are zero? The division ({{math|0/0}}) then becomes meaningless. A case like this is known as an indeterminate form. Other such forms are {{math|∞/∞}}, {{math|0 × ∞}}, {{math|∞ − ∞}}, {{math|0⁰}}, {{math|1^∞}} and {{math|∞⁰}}.}}

{{cite journal |author-first=L. J. |author-last=Paige |title=A note on indeterminate forms |jstor=2307224 |journal=American Mathematical Monthly |volume=61 |issue=3 |date=March 1954 |pages=189–190 |doi=10.2307/2307224}}

{{cite book |author-first1=George F. |author-last1=Carrier |author-first2=Max |author-last2=Krook |author-first3=Carl E. |author-last3=Pearson |title=Functions of a Complex Variable: Theory and Technique |date=2005 |page=15 |isbn=0-89871-595-4 |quote=Since {{math|log(0)}} does not exist, {{math|0^z}} is undefined. For {{math|Re(z) > 0}}, we define it arbitrarily as {{math|0}}.}}

{{cite book |author-first=Mario |author-last=Gonzalez |title=Classical Complex Analysis |publisher=Chapman & Hall |date=1991 |page=56 |isbn=0-8247-8415-4 |quote=For {{math|1=z = 0}}, {{math|w ≠ 0}}, we define {{math|1=0^w = 0}}, while {{math|0⁰}} is not defined.}}

{{cite magazine |author-first=Mark D. |author-last=Meyerson |title=The {{math|x^x}} Spindle |magazine=Mathematics Magazine |volume=69 |number=3 |date=June 1996 |pages=198–206 |doi=10.1080/0025570X.1996.11996428 |quote=... Let's start at {{math|1=x = 0}}. Here {{math|x^x}} is undefined.}}

{{cite book |date=1988 |title=Introduction to analysis of the infinite, Book 1 |author-first=Leonhard |author-last=Euler |author-link=Leonhard Euler |translator-first=J. D. |translator-last=Blanton |publisher=Springer |isbn=978-0-387-96824-7 |url=https://books.google.com/books?id=H58dmcLEnk4C |chapter=Chapter 6, §99 |page=76}}

{{cite book |date=1787 |title=Institutiones calculi differentialis, Vol. 2 |author-first=Leonhard |author-last=Euler |author-link=Leonhard Euler |publisher=Ticini |isbn=978-0-387-96824-7 |url=https://books.google.com/books?id=H58dmcLEnk4C}}

{{cite journal |author-last=Libri |author-first=Guillaume |author-link=Guglielmo Libri Carucci dalla Sommaja |title=Note sur les valeurs de la fonction {{math|0^0x}} |language=fr |journal=Journal für die reine und angewandte Mathematik |date=1830 |volume=1830 |issue=6 |pages=67–72 |doi=10.1515/crll.1830.6.67|s2cid=121706970}}

{{citation |author-first=Augustin-Louis |author-last=Cauchy |author-link=Augustin-Louis Cauchy |title=Cours d'Analyse de l'École Royale Polytechnique |language=fr |date=1821 |pages=65–69 |series=Oeuvres Complètes: 2 |volume=3}}

{{cite book |author-last1=Edwards |author-last2=Penney |date=1994 |title=Calculus |edition=4th |publisher=Prentice-Hall |page=466}}

{{cite book |author-last1=Keedy |author-last2=Bittinger |author-last3=Smith |date=1982 |title=Algebra Two |publisher=Addison-Wesley |page=32}}

{{cite web |url=http://grouper.ieee.org/groups/754/email/msg03270.html |title=More transcendental questions |website=IEEE |archive-url=https://web.archive.org/web/20171114040441/http://grouper.ieee.org/groups/754/email/msg03270.html |archive-date=2017-11-14 |access-date=2019-05-27}} (NB. Beginning of the discussion about the power functions for the revision of the IEEE 754 standard, May 2007.)

{{cite web |url=http://grouper.ieee.org/groups/754/email/msg03292.html |title=Re: A vague specification |website=IEEE |archive-url=https://web.archive.org/web/20171114040300/http://grouper.ieee.org/groups/754/email/msg03292.html |archive-date=2017-11-14 |access-date=2019-05-27}} (NB. Suggestion of variants in the discussion about the power functions for the revision of the IEEE 754 standard, May 2007.)

{{cite report |title=Rationale for International Standard—Programming Languages—C |version=Revision 5.10 |date=April 2003 |url=https://www.open-std.org/jtc1/sc22/wg14/www/C99RationaleV5.10.pdf |page=182}}

{{cite web |url=https://docs.oracle.com/javase/8/docs/api/java/lang/Math.html#pow-double-double- |title=Math (Java Platform SE 8) pow |publisher=Oracle}}

{{cite web |url=https://docs.microsoft.com/en-us/dotnet/api/system.math.pow |title=.NET Framework Class Library Math.Pow Method |publisher=Microsoft}}

{{cite web |url=https://docs.python.org/3/library/stdtypes.html |title=Built-in Types — Python 3.8.1 documentation |access-date=2020-01-25 |quote=Python defines {{mono|1=pow(0, 0)}} and {{mono|1=0 ** 0}} to be {{math|1}}, as is common for programming languages.}}

{{cite web |url=https://docs.python.org/3/library/math.html |title=math — Mathematical functions — Python 3.8.1 documentation |access-date=2020-01-25 |quote=Exceptional cases follow Annex 'F' of the C99 standard as far as possible. In particular, {{mono|1=pow(1.0, x)}} and {{mono|1=pow(x, 0.0)}} always return 1.0, even when {{mono|1=x}} is a zero or a {{mono|1=NaN}}.}}

{{cite web |url=https://www.lua.org/manual/5.3/manual.html#3.4.1 |title=Lua 5.3 Reference Manual |access-date=2019-05-27}}

{{cite web |url=https://perldoc.perl.org/perlop.html#Exponentiation |title=perlop – Exponentiation |access-date=2019-05-27}}

{{cite web |url=https://cran.r-project.org/doc/manuals/r-release/fullrefman.pdf |title=R: A Language and Environment for Statistical Computing – Reference Index |author=The R Core Team |date=2023-06-11 |version=Version 4.3.0 |page=25 |quote=1 ^ y and y ^ 0 are 1, always. |access-date=2019-11-22}}

{{cite web |url=https://doc.sagemath.org/html/en/reference/rings_standard/sage/rings/integer.html |title=Sage 9.2 Reference Manual: Standard Commutative Rings. Elements of the ring Z of integers. |last=The Sage Development Team |date=2020 |access-date=2021-01-21 |quote=For consistency with Python and MPFR, 0^0 is defined to be 1 in Sage.}}

{{cite web |url=https://pari.math.u-bordeaux.fr/pub/pari/manuals/2.11.0/users.pdf |title=Users' Guide to PARI/GP (version 2.11.0) |author=The PARI Group |pages=10,122 |date=2018 |quote=There is also the exponentiation operator ^, when the exponent is of type integer; otherwise, it is considered as a transcendental function. ... If the exponent {{math|{{var|n}}}} is an integer, then exact operations are performed using binary (left-shift) powering techniques. ... If the exponent {{math|{{var|n}}}} is not an integer, powering is treated as the transcendental function {{math|exp({{var|n}} log {{var|x}})}}. |access-date=2018-09-04}}

}}

{{Portal|Mathematics}}

• [http://www.faqs.org/faqs/sci-math-faq/specialnumbers/0to0/ sci.math FAQ: What is {{math|0⁰}}?]
• [http://www.askamathematician.com/?p=4524 What does {{math|0⁰}} (zero to the zeroth power) equal?] on AskAMathematician.com

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