Quote:

"I've spent the last couple of days working on cleaning-up an explanation of these things that makes sense without using a miracle to get to the answer. One of the problems is that there are natural explanations for things like square and cube (area and volume), but, what do powers of 2.1 and 3.25 mean? It is interesting how things completely break down. I don't think I have found a single mathematics text that bridges this gap.

If anyone has a sensible explanation of this I'd love to hear it!"

In other words, I have no use for anything else as it quickly becomes an irrelevant time sink given the stated goal: Trying to explain this to children.

If you can translate what you wrote into something that can be taught to an average teenager (meaning, not a mathematically gifted or advanced student), you might just have the answer.

So far the only explanation I have found for how to solve these kinds of problems is successive approximation by guessing the answer. One level up from there is to use various algorithms to do the guessing, either on paper or through a computational solution (which requires a reasonable level of comfort writing code or using something like Excel).

I do want to thank you for taking the time to contribute to the conversation. Be well.