I've been building a unit & rate library for a forex trading algorithm and I realized I didn't have an answer to this question:
Can currencies be taken to the Nth power?
Unlike physical units, such as length or distance, the square of a currency doesn't occupy a space (I don't have much finance training so there may be theoretical properties that do).
Context:
My library is overloading operators such as $10\text{ USD } \times .98 \frac{\text{Euro}}{\text{USD}}=9.8 \text{ Euro}$. It offers currency and rate classes. If the user wants to square the true value (non unit) they can retrieve the value itself then square that. However, this library is trying to capitalize on relationships between currencies and rates. I plan on expanding it in the future, so I want a strong base. Is there a theoretical and meaningful use of a currency squared?
You can take any value with units (physical such as meters or conventional such as dollars) and raise them to any power--positive, negative, non-integer. It is perfectly legitimate to have a result such as "dollars to the two-thirds." This might be useful for describing some subtle attribute of an economy, say.
However, there is no way you can invest "100 dollars to the two-thirds." In short, the question of what is interpretable or actionable for such non-integer and non-standard units is separate from the mathematical permission to use them.