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Britannica says $a^k$ is the power; Wikipedia says that the power is $k$. Which one is it? Did someone make a mistake?

Exponents

Just as a repeated sum $a + a + \cdots + a$ of $k$ summands is written $ka$, so a repeated product $a × a × \cdots × a$ of $k$ factors is written $a^k$. The number $k$ is called the exponent, and $a$ the base of the power $a^k$.

https://www.britannica.com/science/arithmetic#ref24749

In mathematics, exponentiation is an operation involving two numbers, the base and the exponent or power. Exponentiation is written as $b^n$, where $b$ is the base and $n$ is the power; this is pronounced as "$b$ (raised) to the (power of) $n$".

https://en.wikipedia.org/wiki/Exponentiation

(Bold added in both quoted texts)


I see people insisting that $k$ is a power and $a^k$ is also a power at the same time. While we are at it, why not call $a$ a power too? What do you say?

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    $\begingroup$ Two sentences later in the Wikipedia article: > The exponent is usually shown as a superscript to the right of the base. In that case, $b^n$ is called "$b$ raised to the $n$th power", "$b$ (raised) to the power of $n$", "the $n$th power of $b$", "$b$ to the $n$th power", or most briefly as "$b$ to the $n$th". $\endgroup$
    – Sera Gunn
    Commented Sep 19, 2023 at 2:57
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    $\begingroup$ I am happy saying "$a^k$ is a power of $a$, specifically $a^k$ is the $k$th power of $a$ since $a$ is raised to the power of $k$". But I would call $k$ the exponent rather than the power; it is $a^k$ which is the power. Using th and of makes a difference. $\endgroup$
    – Henry
    Commented Sep 19, 2023 at 13:17
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    $\begingroup$ "Democritus called [the fundamental substance of the universe] atoms. Leibniz called it monads. Luckily the two men never met, or there would have been a very dull argument." (Woody Allen) $\endgroup$ Commented Sep 19, 2023 at 14:24

4 Answers 4

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Both are commonly used: contrary to popular belief, mathematical language is not consistent around the world, nor is it always completely precise. If one says "the power $a^k$" or "the power $n$ in the expression $a^n$", there is no ambiguity and therefore no problem.

And BTW there is no reason to find it embarrassing: that's just the way language is, in mathematics as elsewhere.

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    $\begingroup$ I am not really happy with "the power $n$ in the expression $a^n$" and would prefer "the power of $n$ in the expression $a^n$" $\endgroup$
    – Henry
    Commented Sep 19, 2023 at 13:21
  • $\begingroup$ I upvoted and agree with this answer; however, I prefer to say that the discrepancy in question boils down to definitions (incl. tacit definitions) differing from one another rather than to imprecision. To the OP: which of $7x^5$ and $7(5^x)$ ought to be called a power function? (The former is called a power function and the latter an exponential function.) $\endgroup$
    – ryang
    Commented Sep 19, 2023 at 14:06
  • $\begingroup$ @ryang: I think $a^n$ is the power here $\endgroup$
    – Henry
    Commented Sep 19, 2023 at 14:13
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    $\begingroup$ The mathematical language is precise, it's the English (or other cultural language) that is imprecise here. $\endgroup$
    – rubenvb
    Commented Sep 19, 2023 at 14:16
  • $\begingroup$ What you say would be true, if only those of us speakers of English (or other cultural language) would only agree. Which we probably never will. $\endgroup$
    – Lee Mosher
    Commented Sep 20, 2023 at 0:42
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Welcome to the marvelous world of mathematics and its expression in natural language!

Mathematicians invented formulae to be able to express precise concepts. When you write $a^k$, anybody can understand what you mean, whatever their native language is. A more relevant example if the following formula used in limits definition:

$$ \forall \epsilon>0 \space \exists \alpha>0 , \left | x -x_0 \right | < \alpha \Rightarrow \left | f(x) -f(x_0) \right | < \epsilon $$

It is clear for any mathematician that the value of $\alpha$ depends on the value of $\epsilon$. Depending on the language and the proficiency of both the writer and the reader, a natural language formulation may be ambiguous on that point.

TL/DR: do not expect the natural language expressions for mathematics to be as precise and unambiguous as true mathematics expressions are...

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Wikipedia is wrong, but it doesn't matter.

8 is a power of 2. Specifically, it is the 3rd power of 2. 3 is not a power in this relationship, it is an exponent. You could describe 8 as "2 raised to the 3rd power", or "2 raised to the power of 3", but it would be odd to say it's "2 raised to the power 3".

Of course people are loose with this language, because in reality it almost never causes confusion. If you said 8 is "2 raised to the power 3", people would simply hear it as "2 raised to the power [of] 3" and nobody would be confused. So it really doesn't matter, as others have pointed out.

I would argue that the word "power" becomes a lot less useful when we get out of the realm of integer exponents. When people talk about the "powers of 2", they're usually talking about the set { 2, 4, 8, 16, 32, ... }, possibly including 1, and even less likely including { 1/2, 1/4, 1/8, 1/16, ... }. Nobody would ever call 13 a "power of 2", despite the fact that it is ~$2^{3.700439718141}$. I'd only define $a^k$ as a "power" if $k$ were an integer (although $a$ probably doesn't need to be). 13 is not the "3.7004th power of 2"; it is not a power of 2 at all. But of course, like all language, it depends heavily on the exact context you're using it in.

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    $\begingroup$ +1 for the observation in the final paragraph. $\endgroup$
    – ryang
    Commented Sep 20, 2023 at 6:49
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The terminology itself is consistent. The apparent "mistake" has more to do with purely linguistical aspects.

We may say "function $f$ equals 2", when we in fact mean "[the value of] the function $f$ [evaluated at some point] equals 2". Obviously, a function and a real number are objects of different kinds and cannot be equal. So, technically this is a mistake too. At the same time, we know how to read the sentence so that it makes sense.

Another example is saying "the tensor product of vectors", which may mean one of two things: the operation (i.e. a map from two vector spaces to a third vector space) or the resulting value (for some specified arguments). When dealing with numbers, it is customary to use "multiplication" for the operation and "product" for the value, but the same linguistic rigour is less often applied after high school. It is more important to focus on mathematical rigour than linguistics.

Now back to the question about $a^k$:

  • the base is $a$
  • the power is $k$
  • the whole thing $a^k$ is the [result of evaluation of the] power function;
    therefore, just the power.
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