# Power

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.

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## One thought on “Power”

1. Mike Stay says:

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!