# What is the name of the answer to exponentiation?

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.

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The sum of two numbers is not a product. As for your question, result of an exponentiation is EXPONENT. – Kaster May 16 '14 at 23:49
The sum is a sum – user137794 May 16 '14 at 23:49
Sorry, typographical error. – Ethan May 16 '14 at 23:55
@Kaster Nope. The exponent is the superscript, the little number at the top right corner. The result is a POWER. – bof Mar 9 at 5:36

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 5th 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, ... 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 5th power," rather than "the 5th power of 2," and they tend to think of "5th" by itself as modifying "power" so that 5 is the power, whereas they should think of all of "2 to the 5th" 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.

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

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Neither of these make any sense to me. I thought the power was the exponent. Also, if exponents result in a product, then can't you call the result of multiplication a sum? – Ethan May 16 '14 at 23:57
Both the exponent and the result can be called a power. You can consider multiplication to be the process of repeated addition, so that the result may as well be called a sum. But I don't like that either since it doesn't really hold for $5\times0.7$. I'd just go with power. – user137794 May 17 '14 at 0:35

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 $16$".

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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.

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I believe this is not the case. We have words for difference and quotient, although subtraction and division are not commutative operations. – GOTO 0 Dec 18 '14 at 7:50