I always avoid the word “cancel” in the hope of cutting down on the frequency of this type of mistake, but here I can still see that there’s a -2 being added to a 2, so even with words like “subtracts out” or “adds out” I can still imagine a kid making this mistake.

Someone recently suggested to me having the kids write expressions as trees, so here it would be + branching down to “fifth power” and “zeroth power” and then down to -2 and 2. That would show that the -2 and 2 are not connected by addition. I’d be interested to see a curriculum that shows expressions that way with some frequency. In the meantime I’d be emphasizing that the -2 and 2 have to be connected by addition with no other operations or parentheses getting in the way for you to be allowed to add them up to get 0.

The kid probably freaked when they saw “show a middle step” and had a flashback to 7th grade integers.It’s actually a pretty creative approach.

I know PEMDAS is a dirty word these days, but I think most students would be willing to admit that simplifying exponents comes before addition and subtraction.

This student probably already knows and is comfortable with the idea that any non-zero value raised to the zero power equals one. So unless (-2)^5 = -1, the conclusion needs to be different. Maybe start with number sense before correcting the symbolic mistake.

Trouble is, any z^a + -z^b is not (z+-z)^(a+b)

I’m more concerned that the kid thinks they can add exponents like this. That’s the bigger misconception. But at least they got a common base before misapplying a law of exponents.

I thought I’d let that sit a couple days before providing extra context. It is actually from a test page involving mostly exponent laws, so the speculations of “combining bases as well as exponents” likely isn’t far off. However, in previous examples (not pictured – used other ops), the same base was (correctly) kept, not combined. Also, two other students came up with this error, one assumes thinking similarly.

It kind of floored me, as I’d have usually expected them to keep a “2” and end up with “2^5”, as opposed to adding, then doing exponents. The opposing signs too much of a lure? As to “show a middle step”, it’s so that I know they didn’t just plug the entire expression into the calculator, and the format was the same on a prior quiz, which also included one addition question just to make sure they recognized the difference.

Interesting idea with the trees. Any other thoughts? Context helpful or hinderance?