Talk:Exponentiation

10.1 Exponentiation in Abstract Algebra

From my abstract algebra knowledge of a group, there is no operation called division. Why is it shown here? The proper way to write the inverse is just x^-1 never 1/x, which is not meaningful in an algebraic group. Perhaps this should better be explained in the page, and remove references to 1/x. In general math, yes, 1/x is certainly a valid way to write x^-1, but not in abstract algebra. Conceptually we all handle it the same way, convert 1/x to x^-1 but 1/x is not the operation 1 divided by x. It is the multiplicative inverse of x, that's all. In the group Z3, under multiplication, there is no element 1/2, since there are only elements 0,1,2. However, 1/2 means 2^-1 which is defined, and ultimately 2^-1 = 2. This is easier to explain than the nonsense that results by using the operation division, which would be 1/2 = 2... 24.34.198.111 00:27, 27 April 2007 (UTC)

log (a^b) = b log(a) valid or not valid?

Sorry, I had to correct my question, I have made several mistakes when I wrote it before.

What I meant was:

In the computation of power for two complex numbers a and b as:
a^b
it's used that:
a^b=e^{b log(a)}

But then, in the section Failure of power and logarithm identities, it is said that the identity log(a^b)=b log(a) do not hold in general, and it's prooven with an example.

How is it possible then to equal a^b=e^{b log(a)} , since you need to use the property log(a^b)=b log(a) to write it in that way? What makes that true in this case if a and b are any complex numbers?

Kaexar 01:50, 19 July 2007 (UTC)

What the article says is that $\log(a^{b})$ may be different than $b\log(a)$. For example if a is -i and b is 2, we have $(-i)^{2}=-1$ and $e^{2\log(-i)}=e^{2(-i\pi /2)}=e^{-i\pi }=-1$, as expected. The identity that fails is when you try to take logarithms using principal values, because -iπ is not a legal principal value. — Carl (CBM · talk) 18:32, 18 July 2007 (UTC)

I thank you for your answer but unfortunatelly a parsing error appeared and made half of your post unreadable by the browser :(

But you made me notice that I wrote my question wrong. Thanks again.

Kaexar 01:50, 19 July 2007 (UTC)

Sorry, I have fixed my error. You are right that if you already knew what a^b was then you would need something to rewrite it as e^{b log(a)}. But in the context of complex numbers, this identity is used as the definition of a^b, after the exponential function is defined (via a power series). — Carl (CBM · talk) 01:59, 19 July 2007 (UTC)

power series of e^x

Makholm and CBM. Note that the power series of e^x is not used in this elementary article at all. Nor is the differential equation or the continued fraction. All that stuff is taken care of in the article exponential function. This article on exponentiation is aimed at readers who do not know about exponentiation already. Superfluous material is likely to confuse the reader and make him stop reading. Holes in the argumentation likewise. The power series is a hindrance for reading and understanding. The limit without explanation is a hindrance for reading and understanding. Not everything that is true should be written everywhere. It should serve a purpose. Bo Jacoby 17:16, 8 August 2007 (UTC).

The limit definition of e^x is not used either, except to define e^x. I agree the continued fraction definition isn't needed, but the power series definition is quite important. You will need to justify why you think that the mere inclusion of a formula hinders understanding. Moreover, the "derivation" seemed to add very little to the article. Checking that the derivation is correct would be beyond the ability of the average reader of this article (especially since it used the uncommon notation $\lim _{|n|\to \infty }$ without defining it). We can just give the most important formulas for e^x without "proving" that they are correct. — Carl (CBM · talk) 12:13, 9 August 2007 (UTC)

The limit definition of e^x is used to show definition compatibility. The power series is important, but not here. To many readers a formula is a hindrance, and an unnecessary formula is an unnecessary hindrance. If e^x is defined by the power series the reader will think:

"The most honorable wikipedia editor first defined

e²_{old definition} = e · e

where

e = 1+1+1/2+1/6+1/24+1/120+1/740+...,

and later

e²_{new definition} = 1+2+2+4/3+2/3+4/45+8/315+...

I don't see whether these two definitions provide the same result. I must commit seppuku".

If, on the other hand, e^x is defined by the limit the reader will think:

"The most honorable wikipedia editor first defined

e²_{old definition} = e · e

where

e = lim(1+1/n)ⁿ,

and later

e²_{new definition} = lim(1+2/n)ⁿ = lim(1+2/(2·m))^(2·m) = (lim(1+1/m)^m)² = (lim(1+1/n)ⁿ)² = e² _{old definition} .

I see that the two definitions provide exactly the same result. I'm a genius".

To the service of the reader we should stick to the limit definition. You are welcome to explan |n|→ ∞.

Bo Jacoby 14:15, 9 August 2007 (UTC).

The limit definition is not that much simpler than the power series definition, and the limit calculations are not as elementary as you claim. Readers who don't want to see formulas at all should avoid reading mathematics - we don't need to pander to people who stop reading the first time they come to a formula. Now that the derivation involving the limit is gone, we can replace |n| with the more standard n. — Carl (CBM · talk) 19:05, 10 August 2007 (UTC)

The limit definition is not simpler than the power series definition, but the definition compatibility is much easier to see using the limit definition than using the power series definition. Do you understand the argument above? Of course people reading mathematics should read formulas, but the formulas should serve a purpose, and the power series formula serves no purpose in this context. Why do you want to make these changes, Carl? Bo Jacoby 22:43, 10 August 2007 (UTC).

There is no need to demonstrate the definition's compatibility in this context - we can just assert it. The justification that used to be here for the limit definition was far from accessible to the average person - it required a good understanding of real analysis to transform it into a complete argument. The purpose for including the formulas in this article is to have them accessible if a reader comes to this article to look them up. — Carl (CBM · talk) 22:51, 11 August 2007 (UTC)

By now the article does not even assert compatibility between the two definitions. The main article on exponential function does not explain the compatibility either. The reader is left clueless. The number e is not explicitely defined, and the two formulas for e^x are neither shown to be equal to one another nor to be equal to the old definition of e^x for integer x. So the subsection is useless to the beginner and comprehensible only to the reader to whom it is superfluous. The important formula e^x = lim_n(1−x/n)⁻ⁿ, which before was included as e^x = lim_|n|(1+x/n)ⁿ, has now been omitted, and it is not found in exponential function. The proof for definition compatibility does not really require advanced real (or complex) analysis, because if convergence fails no alternative definition is offered and so no incompatibility issue occurs. You are welcome to include convergence arguments in exponential function, but exponentiation should be kept elementary. Your present edit is a backwards step. I do not question your good faith, but you should take more care. Bo Jacoby 07:59, 13 August 2007 (UTC).

Why do you think that this article "should be kept elementary"? There are no policies and guidelines that imply so. On the contrary, our article about "exponentiation" should say everything that an encyclopedia could be expected to tell about "exponentiation", even if some of it will not be understood by everybody who understands some of the article. We group content by subject, not by levels of sophistication, and there is no requirement that every reader must understand either all of an article or nothing in the article. To keep demanding that is just a wrong editing strategy. –Henning Makholm 09:14, 13 August 2007 (UTC)

See Wikipedia:Make_technical_articles_accessible. Bo Jacoby 06:38, 15 August 2007 (UTC).

That manual of style page reminds us "Do not "dumb-down" the article in order to make it more accessible. " The issue here is not whether to make the article accessible - I'm sure we all have that goal. The issue is more subtle than that. — Carl (CBM · talk) 11:41, 15 August 2007 (UTC)

The name of the article exponentiation is attractive to beginners, (unlike for example Quantum gravity which is more scary), and the article links to the main article, exponential function for readers who want details. The purpose of introducing the exponential function e^x in this article is to prepare the way for the exponentiation a^x. Bo Jacoby 11:30, 13 August 2007 (UTC).

Sorry, it is simply not valid to reason from "the article's title is attractive to beginners" to "the article must not say aynthing except what beginners can understand". That is not how an encyclopedia ought to be organized. The article about exponentiation should tell everything relevant about its subject, not be cut off at some arbitrary level of sophistication. –Henning Makholm 00:14, 16 August 2007 (UTC)

I agree. But the power series for e^x is not relevant for the subject exponentiation. We have exponential function for that. The subsection on integer powers of e should define e and show that the two definitions of e^x for integer x produce the same results. Bo Jacoby 07:53, 16 August 2007 (UTC).

There is a a second reason for defining the exponential function, and that is for defining complex exponentiation. In the context of complex analysis, the power series definition of the exponential function is the important one. — Carl (CBM · talk) 12:52, 13 August 2007 (UTC)

In the context of complex analysis the limit definition e^z = lim_n(1+z/n)ⁿ, is equally important, and the geometrical interpretation of e^i·x = lim_n(1+i·x/n)ⁿ is more straightforward than the geometrical interpretation of e^i·x = Σ_n(i·x)ⁿ/n! Bo Jacoby 19:24, 13 August 2007 (UTC). Note also that the subsection you changed, Exponentiation#Powers_of_e, is part of the section Exponentiation#Exponentiation_with_integer_exponents, where it used to belong but no longer belongs. Please clean it up. Bo Jacoby 06:17, 15 August 2007 (UTC).

I remember that that material was placed with complex exponentiation originally, which I thought made more sense, but you moved it to its current location. The original layout of the article had the first section just on integer exponents of integers. — Carl (CBM · talk) 11:41, 15 August 2007 (UTC)

Do you disagree on the present general layout: integer exponents - powers of positive numbers - powers of complex numbers ? Bo Jacoby 17:06, 15 August 2007 (UTC). By the way, I did not move the power series to the current location, but I removed it, as it is found in many places elsewhere, in e (mathematical constant) and in exponential function and in Taylor's series, and as it is not used in this article. But I didn't find the proof, that the two definitions of eⁿ are compatible, elsewhere in WP, so I included it, and you removed it. I still believe that readers are puzzled by a new definition, without a word of explanation, of a concept that has just been defined. Don't you? Bo Jacoby 07:44, 17 August 2007 (UTC).

If we give two definitions and just say "they're equivalent", readers will believe us. If they wonder why they are equivalent, they can look at Characterizations of the exponential function, linked from exponential function. — Carl (CBM · talk) 01:55, 20 August 2007 (UTC)

Exponentiation has defined a^k for integer k and now defines e^k for integer k without saying (or showing): "they're equivalent" . It's not about lim_n(1+x/n)ⁿ = Σ_nxⁿ/n! for real or complex x, but about lim_n(1+k/n)ⁿ = (lim_n(1+1/n)ⁿ)^k = e^k for integer k . Characterizations of the exponential function does not show it. Nor does e (mathematical constant). Bo Jacoby 06:31, 20 August 2007 (UTC).

You have your priorities reversed. In the context of complex exponentiation, one first defines e^x, and then defines e as e^1. One does not first define e and then define e^x from it. The point is that e^x is very straightforward to define and prove analytic, unlike 2^x or π^x. — Carl (CBM · talk) 14:04, 20 August 2007 (UTC)

One can define a function exp(x) . Without saying (or showing) that exp(1)^k = exp(k) for integer k, the notation e^x is unjustified, and it makes no sense to the reader to include the exponential function in exponentiation . Bo Jacoby 09:09, 21 August 2007 (UTC).

Principal n^th root

In " If a is a real number, and n is a positive integer, then the unique real solution with the same sign as a to the equation \ x^n = a is called the principal n^th root of a, and is denoted \sqrt[n]{a}."

When you say "the unique solution" is it not implicit that there always is one? (not the case!!) And why use the "the same sign"? (what about a=0??) Wouldn't it be more clear/didactic to explain what happens dividing in cases, the n's into even or odd and the a's into positive, 0, and negative?

Why use the term "principal n^th root"? It only makes real sense after someone studied complex numbers, which still takes a few years for most students. Ricardo sandoval 04:28, 24 August 2007 (UTC)

The principal nth root is unique. The nth root is not always unique, even when operand and results are real numbers. The sentence would not be correct without "principal". Principal does not refer to complex numbers. See n-th root or square root. About "with the same sign as a", in the case that a=0, n=0, and the convention 0⁰ = 1 is adopted, you are right... In all the other cases, the expression "with the same sign as a" is correct. The special case can be briefly described later as an exception (but this should be done also in the article about n-th root).

It is indispensable to explain in this article the principal nth root (because it introduces the rational powers). It is not indispensable to explain the nth root. Before my 21 August edit, the authors only described the simplest case of principal square root, with positive a, n and x, and they did neither call this "pricipal n-th root" nor "one of the two possible n-th roots", but simply and incorrectly "the n-th root".

However, there's a separate article about n-th root, and the internal link is now clearly specified on top of the section... Paolo.dL 09:53, 24 August 2007 (UTC)

I was pointing to the absence of principal even root of negative numbers as in n-th root. So the term "the unique" doesn't apply. Someone could define the principal nth root for these number but they would not be real anyway.

I see the text is an improvement over the previous one (to which you referred) and it is even better now. But there is still the problem above. When remarking about the nth root of zero, I was pointing to the more trivial fact that zero has no sign, in the usual sense ( I am not considering the sgn function as a definition of sign). Although someone should expect the principal nth root of someone "without sign" (0) should also be "without sign" (0), this is not stated explicitly. Ricardo sandoval 03:43, 25 August 2007 (UTC)

Well, the sentence about the n-th root of negative numbers (copied from the lead of the article n-th root), was already included at the end of the section. About the n-th root of zero, it's a very special case, and it is easily understandable that "zero has the same sign as zero", even if zero has no sign. This for sure will not create misunderstandings or ambiguity. I don't see the need to say it explicitly here, in a section that is a summary of a clearly specified "main article", which the reader can easily check in case of doubt. Paolo.dL 10:17, 25 August 2007 (UTC)

In the section Rational powers of positive real numbers it is implicit that the result is also positive, i.e that the negative root should not be taken, or the principal value should be used. I think that should be made explicit. —Preceding unsigned comment added by TerryM--re (talk • contribs) 00:11, 4 November 2007 (UTC)

GA on hold

This article has been reviewed as part of Wikipedia:WikiProject Good articles/Project quality task force. In reviewing the article against the Good article criteria, I have found there are some issues that need to be addressed. The article is under-referenced. Also, it's not specific enough for stating references in general. I am giving seven days for improvements to be made. If issues are addressed, the article will remain listed as a Good article. Otherwise, it will be delisted. If improved after it has been delisted, it may be nominated at WP:GAC. Feel free to drop a message on my talk page if you have any questions. Regards, OhanaUnited^{Talk page} 21:32, 8 September 2007 (UTC)

Whoever can take care of the images that display equations, check out the first "Justification" under the heading "Zero to the zero power." It claims X^X -> 1, where it should be X^0 NinjaSkitch 06:06, 1 October 2007 (UTC)

"a to the power n"?

I realize that "a to the power n" is correct terminology, but there is another correct notation that could be included there, "a to the exponent n". My math teacher tells us that that is the correct form, and that "a to the power n" is incorrect. However, I don't believe her, and I think both are correct. Is my math teacher wrong, or is she right? ZtObOr 01:42, 9 October 2007 (UTC)

EDIT: I forgot to include this bit. My math teacher's reasoning is that since the "n" in "a to the power n" is called the exponent, that it should really be called "a to the exponent n", since the word "exponent" refers to the exponent "n" in the first place, even if the word "power" is in its place.. —Preceding unsigned comment added by Ztobor (talk • contribs) 01:45, 9 October 2007 (UTC) (Edited 4/10/2008)

It's better not to argue with your math teacher. — Carl (CBM · talk) 02:34, 9 October 2007 (UTC)

Of course, if my math teacher is indeed wrong, then it should be okay to argue with him. (I'm in a different grade now, and we have a male math teacher now.) ZtObOr 02:04, 5 October 2008 (UTC)

Perhaps it is too pedantic to say that "a to the power n" is not a notation; it is a verbal description, or terminology. The notation is $a^{n}$. As to the answer to your question: $n$ is the exponent, as your teacher rightly says. It is also the power. The exponent is the symbol that we use to designate a power. Therefore I prefer to say "a to the power n" as it doesn't confuse the thing with the symbol that we use to describe it. This is rather like the distinction between a number and a numeral. Of course you may choose to finesse the problem by saying "a to the n".TerryM--re 00:01, 4 November 2007 (UTC) I should add that the term index is also almost synonymous with exponent. In fact now that I think about it, I'm not sure that the terms are well defined; they are certainly used almost interchangeably by many people.TerryM--re 00:31, 4 November 2007 (UTC)

I don't get your saying n is both the "exponent" and the "power". If n is the exponent, doesn't that preclude it from being the entire expression?

But you do have a point. I say "a to the power n" because the word "power" refers to the entire expression, not just "n". If I were to say "a to the exponent n", that would mean that aⁿ is the exponent, and not just n. ZtObOr 01:15, 6 October 2008 (UTC)

In most mathematical contexts, "index" means the number that identifies one of several variables that have been given names from a numbered sequence. An index is often notated as a small lowered number following the variable letter, whereas an exponent is notated as a small raised numbers. For example, $a_{3}x^{5}$ means a-with-the-index-3 multiplied by x to the fifth power. In tensor calculus some indices are written as raised numbers, rather than lowered ones, but that is a specialty usage. –Henning Makholm 01:58, 4 November 2007 (UTC)

The index can also refer to the "index" of a radical, or the exponent by which you reverse the operation by. ZtObOr 01:56, 5 October 2008 (UTC)

I have never seen the phrase "a to the n-th" ever used in any literature, nor have I every heard this phrase being used. If one says "a to the n-th" without "power" following, what does this mean? For example, if I say, "a to the fifth," do I mean a⁵ or a^1/5? European and American usage differ here, and "a to the n-th" is, to my mind, ambiguous. Xantharius (talk) 16:03, 6 October 2008 (UTC)

Notation: cis (x) = cos (x) + i * sin(x) = e^(ix)

Is there any article to link to, to mention the cis function? This function can make working with Euler and De Moivre's formulas a lot easier. It is clearer to write on paper and leads to less mistakes, not to mention faster. An online example of cis, although not specific: http://oakroadsystems.com/twt/sumdiff.htm

Also, a link to an article describing implementing complex exponention in software: http://www.efg2.com/Lab/Mathematics/Complex/Numbers.htm The code is in Pascal/Delphi. JWhiteheadcc 06:05, 2 December 2007 (UTC)

I do not think the name "cis" is standard or widespread, except perhaps as a didactic device used when teaching the complex exponential to beginning students. The function itself is often needed in many applications (quantum physics, AC electronics, higher algebra, to name a few), but there it is invariably written e^ix which is shorter and eliminates the need to remember a different notation for the special case of x being real. Euler's formula is already in the article, but perhaps deserves to be displayed. –Henning Makholm 12:25, 5 December 2007 (UTC)

I have the impression that 'cis' is more common in engineering. — Carl (CBM · talk) 13:43, 5 December 2007 (UTC)

OK; I don't have many engineering texts to compare with. –Henning Makholm 17:41, 5 December 2007 (UTC)

Powers with infinity

This section, when I came upon it, claimed that Cantor's theorem implies that $\infty <2^{\infty }$ in the context of calculus. This is nonsense. If $2^{\infty }$ is interpreted as the limit of 2^g(x) as x goes to infinity (where g(x) is a real valued function which goes to infinity), then this $2^{\infty }$ is just a limit of real numbers and not some kind of cardinal which is "larger" than another infinite limit of real numbers. Not larger in the sense of Cantor's theorem, anyway. This $\infty$ is not a cardinal.

So I fixed up the section so that it at least says correct things instead of nonsense, but I don't think it's a great section. I wouldn't mind if someone just deleted it, but then someone else would probably repost the same misconceptions later on.

By the way, I don't know where on earth I could find citations for this kind of thing. It's clear to any mathematician familiar with the relevant definitions. —Preceding unsigned comment added by 70.245.113.97 (talk) 01:43, 11 December 2007 (UTC)

Dismissing Cantor's theorem as nonsense is itself nonsense. The text as it was, was a necessary counterbalance to the way people are flinging around infinity in the section. If you want to say it makes no sense from the POV of calculus then say it, don't delete the fact that it makes complete sense from a set POV. --Michael C. Price ^talk 06:43, 28 August 2008 (UTC)

Michael, really, that text doesn't make sense, and I don't think 70.245 was claiming there was anything nonsensical about the theorem, just about its application here, which genuinely is out of place. The symbol ∞ is not used for transfinite cardinals. I'm going to assert that flatly; it is simply not used that way, at all, in set-theoretic publications. (It is used, occasionally, for the notional "cardinality" of the collection of all ordinals, but that's so far from being a reasonable reading of the symbol in the current context that it doesn't even come up here.)

The text cannot stand as it is. My only doubt is whether to remove that paragraph alone, or to cut deeper, maybe deferring the whole section to extended real number line, which is where this discussion really belongs. --Trovatore (talk) 08:11, 28 August 2008 (UTC)

Then perhaps there should be a section about exponentiating transfinites or alephs or whatever. Eiether way the subjects should be mentioned somewhere in this article. --Michael C. Price ^talk 08:43, 28 August 2008 (UTC)

There's already such a section: Exponentiation of cardinal and ordinal numbers. --Zundark (talk) 11:54, 28 August 2008 (UTC)

Which nowhere mentions Cantor's theorem, nor transfinites, aleph etc. If the article is really going to have misleading statements such as 7^∞ = ∞ in it, then let's see some citations.... --Michael C. Price ^talk 12:14, 28 August 2008 (UTC)

The section on exponentiation of cardinal and ordinal numbers doesn't mention Cantor's Theorem, but it's where Cantor's Theorem should be mentioned if this article is going to mention it at all. The paragraph on cardinal exponentiation contains little more than a definition, and the definition doesn't depend on whether the cardinals are finite or infinite (= transfinite = an aleph). More details are in the Cardinal exponentiation section of the cardinal number article. As for 7^∞ = ∞ (which is apparently intended as a shorthand for 7ⁿ → ∞ as n → ∞), I wouldn't object if someone simply removed the entire "Powers with infinity" section. --Zundark (talk) 12:47, 28 August 2008 (UTC)

I'm inclined to agree.--Michael C. Price ^talk 14:26, 28 August 2008 (UTC)

I agree too; things like 7^∞ are just poor notation, as is most of the rest in that section. Shreevatsa (talk) 15:47, 28 August 2008 (UTC)

Am I right in thinking that $n^{\infty }$ tends to $\infty$ when n>1, remains at 1 when n=1, tends to 0 when abs(n)<1, and tends to $\pm \infty$ when n<-1? Note: This being my first contribution, I don't want to make changes directly, in case of misinterpretation of the intent of the article! MessyBlob (talk) 18:48, 5 April 2008 (UTC)

What would be correct to say is that n^k tends to ∞ as an integer k tends to ∞ when n > 1; tends to 1 when n = 1; and tends to 0 when abs(n) < 1. The expression n^k cannot converge to ±∞ if n ≤ −1, because the notion of convergence means that it gets close to one value, not more than one. In this case it would therefore probably be okay to write n^∞ = ∞ if n > 1, that n^∞ = 0 if abs(n) < 1, that n^∞ = 1 if n = 1; but note that n^x might not even be a real number if n is negative, and I do not immediately see any reasonable way to define n^∞ if n ≤ −1, because the value of n^k oscillates between 1 and −1 if n = −1, and between increasingly divergent positive and negative numbers if n < −1. Xantharius (talk) 19:50, 5 April 2008 (UTC)

Agree that convergent and divergent series is the way to go on this. It might be useful to express the case of n = −1 in terms of a wave in the imaginary axis, using Euler's equations as basis. For n < −1, the same would apply, but tending to infinite amiplitude. You could probably reach intermediate values for n =−1 using a cosine function of phase, like $-1^{\phi }=\cos {\pi \phi }$. —Preceding unsigned comment added by MessyBlob (talk • contribs) 17:12, 6 April 2008 (UTC)

0 to power of 0

In this section, it is heavily implied that the Microsoft Calculator is a programming language, which is obviously wrong. Maybe it should be kept for simplicity or perhaps a rewording of the statement is needed. —Preceding unsigned comment added by 88.108.69.114 (talk) 19:37, 17 May 2008 (UTC)

I edited the text to fix this, and added reference to Microsoft Excel. Paolo.dL (talk) 16:06, 18 May 2008 (UTC)

the only way i see this = 1 is the ^0 is subtracting 1 less so 0^0 = 0-(-1)= 1 —Preceding unsigned comment added by 208.91.185.12 (talk) 21:31, 2 October 2008 (UTC)

0 to the power 0: Knuth

I'm surprised the 0⁰ section does not mention all that Knuth has said in his paper Two Notes on Notation. Can someone add it? The paper: TeX source, doi:10.2307/2325085, arXiv:math/9205211.

Possible error in section 1.5

I'm not a math person, but I think I see an error in section 1.5 Identities and properties and would like someone help me check this out.

In the last line of the section the statement is a^b^c = a^(b^c), but the example preceding this seems to show just the opposite.

Then there is an inequality a^b^c ≠ (a^b)^c. Should this be an equality not an inequality?

And, finally, the article states "Without parentheses to modify the order of calculation, the order is usually understood to be from right to left". Should this be "from left to right"?

```` —Preceding unsigned comment added by Lou27182 (talk • contribs) 16:14, 27 July 2008 (UTC)

a<sup>bc</sup> means a<sup>(bc)</sup>, and (a<sup>b</sup>)<sup>c</sup> is written a<sup>bc</sup>. Bo Jacoby (talk) 16:32, 27 July 2008 (UTC).