I am working on a software library for Units of Measure. To represent dimensions, I need to know the required range of exponents for each of the seven base units (precisely, I need to know the required maximum number of bits for each exponent). After reading many related questions, I still have some questions.
As far as I understand, all physical quantities can be expressed as the product of the seven base units, with each base unit raised to the power of $0$, $\pm1$, $\pm2$, $\pm3$, and so on. So the dimension of a quantity is represented by a vector / array of 7 integers.
E.g. for volume (length ^ 3) that array would be (3, 0, 0, 0, 0, 0, 0), for acceleration (length / (time ^ 2)) it would be (1, -2, 0, 0, 0, 0, 0).
While I could calculate the square of a volume (with dimension 6,0,0,..), it doesn't make much sense, because I have never encountered any physical quantity that needs (length ^ 6).
a) What is the maximum required exponent for Length? Do you know any example where $\pm3$ is not enough?
b) Maximum exponent for Time? Any example beyond $\pm4$? (update: raised to 4 for capacitance)
c) Maximum exponent for Mass? Any example beyond $\pm1$?
d) Maximum exponent for Electric Current? Any example beyond $\pm1$?
e) Maximum exponent for Temperature? Any example beyond $\pm4$? (update: raised to 4 for Stefan-Boltzmann constant)
f) Maximum exponent for Amount of Substance? Any example beyond $\pm1$?
g) Maximum exponent for Luminous Intensity? Any example beyond $\pm1$?
... plus three bonus questions:
h) Is there some required order of the seven base units? I have seen more than one ordering in use.
i) Are those seven base units enough for everything, or is there some bleeding-edge science that needs more?
j) Are the exponents always integral? Never ever something like ^ 1.5 or ^e?
I guess I should clarify: I am aware that values might need rational or real exponents. I am aware that intermediate results might need a wider range of integer exponents. I have read the relevant SI standard document, and lots of websites about the topic.
My questions are just about the base units exponents that define the dimension of (base or derived) units.
The most precise statement I found so far was: the exponents are very small integers.
However, to exploit all filthy speed tricks of CPU or GPU assembly programming, I need to know the exact number of bits required.