Iaman Does Algebra

Subtract the exponents when dividing variables.

Iaman is studying algebra:

I was wondering about the difficulty of learning Algebra,  and as importantly having it stick.
I went through 501 algebra problems, helped with a answer key, finally getting comfortable with  how  things worked.  But then was uncomfortable when doing on my own, without a key.
I am not the only one, googling shows many studies showing the following.

    "Our results indicate that the transition from concrete to abstract reasoning represents quite a long process, even for simple algebraic task used in this study. ............be aware that it is not enough to learn about equation rearrangement in mathematics once.
     It should not be presumed that students master this skill quickly and that they can easily apply it in other context such as problem solving in physics.
   On the contrary, teachers should use every opportunity to encourage students to use formal reasoning – both pattern recognition and effective application of mathematical rules and known procedures."
From Development of abstract mathematical reasoning: the case of algebra

    I was always good at math*, and I am occasionally complimented on my ability to explain semi-complicated things to other people. However, there was one occasion that baffled me. A friend of my kids was having trouble with high school algebra, and I was volunteered to help. I tried to explain the concept (the distributive property, maybe?). I went over it three or four times, but we weren't getting anywhere. I had exhausted my repertoire of techniques, so I gave up. Thing is, this kid was one of the brighter ones, they went on to college and professional career. Maybe there are other techniques of which I was unaware, or maybe their brain just wasn't ready for abstract thinking.

* That is, until we got to differential equations and then the requirements changed from understanding the concepts to rote memorization of complex formulas. That sudden reversal really turned me off. Only recently have I come to realize that every one of those complex equations probably represented someone's life work. However, at the time, no one bothered to mention that.