What is the name of the answer to exponentiation?

Question

What is the name of the answer to exponentiation? Adding two numbers produces a sum. Multiplying two numbers produces a product, but I cannot think of or find the name for the solution to exponentiation.

Answer 1

According to Wikipedia, the result can be called a power or a product.

Answer 2

The result of exponentiation is called an the $y$th power of $x$. As an example, one would say, "The $4$th power of $2$ is $16$".

Answer 3

Note that unlike the case of addition and multiplication, the binary operation of exponentiation is not symmetric in its arguments. There can't be just one word denoting the result of applying exponentiation to a pair of numbers.

For example, if I gave ou the problem of applying the exponentiation operation to the pair of numbers $2$ and $3$ and that was all the information I provided you, you'd have no way of knowing whether I meant $2^3=8$ or $3^2=9$. The reason words like "sum" and "product" exist is because of the commutative properties of addition and multiplication. If addition and multiplication weren't commutative, we'd just say $a$ plus $b$ or $b$ plus $a$ and the like to refer to the result of the operation and leave it at that. But since no matter what order you add numbers you always the same value, it makes sense to create a term referring to that value, e.g., 'sum'. Exponentiation just isn't analogous to addition and multiplication in this respect.

Answer 4

The correct answer is power.

In an expression like $b^x$, $b$ is called the base, $x$ is most commonly called the exponent but sometimes called the index (actually power is also commonly used, but erroneously), and the overall result is called the power.

One can say, "the $5$th power of $2$ is $32$." What is $32$ then? It is a power, specifically the fifth power of $2$. We talk about powers of $2$ (or other bases), such as $1, 2, 4, 8, 16, \ldots$ Note that $3$ is not a power of $2$, so if one sees $2^3$, $3$ should not be thought of as a power. Unfortunately, people get sloppy in their verbal expressions and might refer to "$2$ to the $5$th power," rather than "the $5$th power of $2$," and they tend to think of "$5$th" by itself as modifying "power" so that $5$ is the power, whereas they should think of all of "$2$ to the $5$th" as what is modifying "power".

This potential backwardness is not unique to powers but applies also to division. We can say "$3$ divides $12$ four times" or "$12$ divided by $3$ is $4$"; in the former case the divisor is stated first whereas in the latter case the dividend is stated first.

The bottom line is that we do not need to have power serve as a synonym for two already existing terms (exponent and index), while we are needing to have a name for the result of the operation.

Answer 5

I agree that calling both the exponent and answer power is a bad idea. We need a more specific name. Already we can call any result from any operation " a result, answer, number, sum, output, yield, yielding answer/result...etc.".

The result of powering is called the power. The confusion lies in the ambiguity. When I say "the power" am I referring to the exponent or the answer? Product seems safest because it is already well defined in terms of multiplication and all we are doing is repeated multiplication of a base number.

We have the base (or radix), the exponent (or index or power) and the power (or product). The fear is whatever unique name we try to call the answer will also became a name for the exponent. We should choose a name that is unique to the answer and forbidden from being used as a name for the exponent or else it defeats the point.

A power is a square or cube or higher dimensional entity. We should call the exponent the dimension and the answer the "entity". Entity is hands down the best choice when used in this context as it means "thing" and thing is defined as " I can not give a name to this but would like to".

The radix is the Latin root forming "radical". The we could take index and turn it into "indical" as it is referring to the exponent but not the exponent itself hence saying it is the "power" of the number represented as an answer but not "power" as an exponent.

Word play using synonyms for "object", "shape" or "number" do not produce as meaningful new words. Using a funny sounding make up word like "indical" you can see the connection to the index and hence the power but it sounds too funny to use for anything thing else than the answer. As such it would only be used when you need to specifically refer to the answer. It sounds legit enough to gain acceptance while being funny enough that most people will just say "product" to avoid confusion with the exponent.

Your best choices are entity, indical, power, product, in that order. Your worst choices are power or a name stemming from shape. I tried to use nary or n-ary or some change of n-tuple. Exponent-tuple = e-tuple? It is almost there but not simple enough and that is what the poster is asking for. I tried using modified versions of rank or ordinal but they were too confusing, defined elsewhere, and/or they can be used for the exponent itself. While this is not a bad thing it is something that lacks specificity and I wished to avoid.

We can "square or cube" a number so a generic form for any higher dimensional shape could work but the simplest and easiest to accept is "entity" which means thing or dimension. I vote for entity, product, or indical.

Edit: It should be called the proponent. Exponet is latin, "ex" means outside and "ponent" means position. We can take this to mean that the exponent is outside of the base number, and proponent is after the base number being on the other side of the equal sign. The word component was considered but lost to entity since a component is defined as being part of an entity. Component is also used in vector sums and wished to avoid dual meaning if possible. Proponent means before or after a position. When we are taking about base we are talking about a positional system of where the digits are. 3^2 = 9 has 3 as the base, 2 outside of the base and 9 after the base.