Our language surrounding exponents is confusing and, I think, misleading.
Power
An exponential relationship involves three values. Historically, these were called the base, the exponent, and the power.
On a logarithmic scale, the base represents the step size, the power represents the target value, and the exponent represents the number of steps. For instance, if you measure the distance from 1 to 5, and triple that length, you wind up at 125.
There are three ways to represent this relationship: \[5^3 = 125 \Leftrightarrow \log_5 125 = 3 \ \Leftrightarrow \sqrt[3]{125} = 5\]
Consider \(\sqrt[3]{125}\): We say “the third root of 125 is 5” or “the cube root of 125 is 5”. It is clear what the root is: It’s five.
Consider \(5^3 = 125\), though. While we say “the third power of 5 is 125”, we also say “5 cubed is 125”, “5 to the third is 125”, “5 raised to the third power is 125”, and several other constructs.
This has led many people, including teachers, to believe that the power is the exponent, and so that language is shifting. This is a disappointing shift: Each part should have a clear name. “Power” should clearly refer to either the exponent (which already has a name) or the result of exponentiation (which otherwise has no name), but not both depending on the person.
I believe the confusion comes from the notation and on our temptation to read from left to right. Since the three is to the left of the 125 in \(\sqrt[3]{125}\), it’s natural to read that as “the third root of 125”; however, since the three is to the right of the five in \(5^3\), it’s less natural to read that as “the third power of 125”, and so the temptation is to read it as “five to the third power”.
Another reason for the confusion may be the sort of questions we ask (and don’t ask) in algebra: It’s common to write \(5^(2x+1) = 17\) as an exercise in logarithms, where “the 2x + 1st power of five” is an awkward thing to say. In contrast, students may never see an exercise like \(\sqrt[2x+1]{56} = 12\).
Indeed, overall, we tend to keep radicals to those with natural number indices greater than one, so even \(\sqrt[0]5 = 1\) and \(\sqrt[1]5 = 5\) are unusual things to write (even though they’re mathematically accurate). This is why the default index for radicals is two.
Raised to
To me, “raised to” implies an increase. Even so, I hear and read people using this language even when the exponent is less than one. When we read \(125^\frac13\) as “125 raised to the one-third power”, what does that even mean? What about reading \(4^{-2}\) as “4 raised to the negative two power”? In both cases, the result is less than the base.
I sense that the origin of this language is because we write exponents as superscripts (and I don’t think we should, but that’s another topic). But does it make sense to have our notation drive our language? I’ve seen even traditionalists who complain about reading 3.54 as “three point five four” (preferring “three and fifty-four hundredths”) who then use “4 raised to the negative two”.
The other way that “raised to” language would make sense is if we consistently saw exponential relationships in terms of a vertical logarithmic scale starting at one, where numbers are then found by lifting a metaphoric elevator to a specific location, but even then, negative exponents would result in lowering (from one), not raising.
I understand we want a verb of some sort (although we don’t seem to want for such a verb for logarithms or roots). We have verbs for the other basic operators: \(4 + 5\) can be either “four plus five” or “four added to five” (or “five added to four”); we subtract; we multiply; we divide. So it makes sense that we want to “raise”. And an argument could be made that multiplying and adding both imply an increase, but it doesn’t feel as focal to me as it does for “raise”.
Alternatives
So… what to use instead?
The thing is: I don’t know. I feel like this language is so firmly entrenched that it’s a losing battle to fight it. This seems especially true of calling exponents “powers”, although I’m pushing myself, at least, to avoid that.
We could leverage the idea that sound is measured logarithmically and use “amplified”, but a form of that is already used for linear amplification (“amplitude”).
We could use “shifted” (for the motion along the log scale) or “aggregated” (for the concept), but those both feel vague.
And do we even need a verb? We don’t have a common verb (to my knowledge) for logs or radicals.
One problem, I think, does go back to the awkwardness of “the [nth] power of…” when n is not a positive integer. And it also strikes me as misleading to read \(5^3\) as “five with an exponent of three” because it’s still five; the value of five is not changing. We’re referring to the third power of five, not to five itself.
Ultimately, I think the best route would be to get over the awkwardness of “the [nth] power of…” because that’s at least accurate, while the other ways of describing it are somewhere between misleading and downright false.
At the same time, I realize that mathematics communication is a language and hence is not going to be completely logical. Which is to say, I don’t expect to win this issue, but at least I can voice my ongoing objections.