I know the pic is a bit small, but can you see the mistake? It all has to do with what the exponent applies to. Somewhere on the internet one of you wrote about how you tell kids that “the exponent only sticks to one thing.” This mistake is about just that.

Thanks to Gregory for the submission.

• I told my kids it only applies where it is immediately adjacent, but they still made this mistake. I like the stickiness concept. It would be interesting to see if they would make the same mistake if you wrote it 5x^2(x^3)^-4. In other words, is the first set of parentheses triggering that association thing so that they think the exponent applies there, too?

• I’ve said something similar – be aware of your base – yet a quarter of the class made the error (I just scanned in six, including the most interesting ones). The parentheses argument is possible, except many wrote in a valid middle step with no parentheses anywhere.

The comment does make me wonder, would using 5^2 (or y?) cause them to put 25 (or y) in the denominator? It’s like coefficients are “just along for the ride”. The stickiness idea might help with the separation; I haven’t tried that language. One can hope!

• When I introduce and reteach, reteach, and reteach, I show each individual piece as separate factors; then urge them to deal with each individual piece: (2x^2)/(4x^-3) would be 2 * x^2 * 1/4 * 1/(x^-3). They can then rearrange physically to multiply the most logical factors together first.