As my daughters learned about exponents, they waded through a sea of definitions and formulae. I taught exponents in algebra class, and it really surprised me that exponents were such a difficult topic. There are only a few rules to remember; it seemed straightforward to me. Rules about when to add exponents, when to multiply them, and when to change their sign.
But this is after years of being a mathematician. It’s not that the exponent rules are so ingrained in me that I never make a mistake simplifying expressions. It’s that exponents have a big set of associations for me. I recognize them in different guises. Much like my brother’s face.
For a long time, my brother’s face looked like this:
Now he looks rather different.
But the nice thing about knowing someone for a long time, and through many changes, for me anyway, is that all the essentials are still there. To me his gray hair is not his basic hair color. It’s a recent dusting of frost. He’s still the same. It’s the same with all the faces in my family.
What I’m saying is: when you first learn about exponents, your understanding is like my brother’s face above. Your understanding will grow, and change, and become unforgettable and recognizable in many guises. You might still mistake that face once in a while. But with time you can know it, very well.