I brought this up once before and it triggered a reminder about another subject we put too much importance on in Algebra II: exponent rules. Again, they are important and they have a place, but sometimes it just goes a little too far:
Similar to the other exercises I pointed out, these appear towards the end of a problem set. But what do these represent? Can you name something that is modeled by a ninth power? How often are scenarios controlled by 3 variables of incredibly high powers like this?
I know what you're saying and I agree, the ability to simplify expressions like these serves to demonstrate a high level of comfort with mathematics. Too often I've taught this unit and the exponent rules only serve to confuse. Students are very good at subtracting or adding the powers of like variables, but they like to apply it to the constant as well. In #32 for example, a good number of my students would put an 11 in the denominator of the simplified answer.
Is that a failure on my part? Sure. Should we spend more time on the underlying misconceptions? Sure. Do we have the time to do that?
That last question is the tricky part. The length of Algebra II curriculum is so long that there is pressure to keep the foot on the gas. With a long parade of functions to get through, we've 3 days to master these exponent rules kids and that's it. The problem is they never get mastered because there's no other areas of Algebra II where it's necessary to reduce 8th, 9th, and 10th powers. It'll appear in Calculus, but the context of finding the derivative of an 8th power is just as contrived.
I also hear you say "but students are being forced into remedial math now more than ever!" Will continuing to power through an incredibly long curriculum help this? Or did the curriculum creep cause this in the first place? Would a more focused topic list in high school drive home low-level skills that would improve scores on college math placement tests?
And speaking of contrived context:
Not an authentic textbook problem, but we've all seen something like this in an attempt to work in a geometry crossover.