I have rewritten the entry to “power” in the abstractmath.org Glossary:

## POWER

Here are three variant phrases that say that $125=5^3$:

- “$125$ is a power of $5$ with exponent $3$”.
- “$125$ is the third power of $5$”.
- “$125$ is $5$ to the third power”.

Some students are confused by such statements, and conclude that $3$ is the “power”. This usage appears in print in Wikipedia in its entry on Exponentiation (as it was on 22 November 2016):

—

“…$b^n$ is the product of multiplying $n$ bases:

\[b^n = \underbrace{b \times \cdots \times b}_n\]

In that case, $b^n$ is called the $n$-th power of $b$, or $b$ raised to the power $n$.”

—

As a result, students (and many mathematicians) refer to $n$ as the “power” in any expression of the form “$a^n$”. The number $n$ should be called the “exponent”. The word “power” should refer only to the result $a^n$. I know mathematical terminology is pretty chaotic, but it is silly to refer both to $n$ and to $a^n$ as the “power”.

Almost as silly as using $(a,b)$ to refer to an open interval, an ordered pair and the GCD. (See The notation $(a,b)$.)

**Suggestion for lexicographical research:** How widespread does referring to $n$ as the “power” come up in math textbooks or papers? (See usage.)

Thanks to Tomaz Cedilnik for comments on the first version of this entry.

There are, of course, other uses of power in mathematics, like https://ncatlab.org/nlab/show/power, but anyone likely to encounter that meaning will have no trouble with the one you describe above!