This post by Michael Pershan has an excellent way to give students opportunities to play around with exponents and problem solve with them.
The basic idea is, give students a bunch of dots in an array, and ask them to find ways to change the picture so that it represents a power.
A related, and more generic, question is:
How do I change this diagram/image/equation/formula so that it represents <something else>?
Related but slightly different question I’ve tried to get at with some number talks on this topic — how does changing the diagram help us learn more about it?