Actually, the priors have been introduced in an earlier chapter, so I should mention that. I talk about the rule of subtracting exponents when dividing earlier. So what you describe has already been presented.
This is all good. I think my own approach would differ slightly, but it might require more work later -- at least in principle -- when one moves on to non-integer exponents. I'd probably start with establishing the obvious extension of equivalent fractions from positive integer numerator/denominator to arbitrary or unspecified values x,a,b: (xa)/(xb) = a/b (x≠0). Then I'd do powers a few like x^2/x^1 = (xx)/(1x) = x/1 = x and x^2/x^3 = (xx)/(xxx) = (x)/(xx) = 1/x = x^(-1) etc and eventually extend to a rule via iteration that x^a/x^b = x^(a-b) when a≥b and 1/x^(b-a) when a≤b, then observe that both may be expressed in the form x^(a-b) when allowing negative exponents. As always what one can (or maybe one should say "should") or cannot do really depends on priors: what have students mastered before the lesson/unit begins? I like the principle that a mathematical problem I solve in the present becomes a tool I use in the future, which is, I know, anathema to the progressive framework of math ed (hey but that's *only one* of its merits :-) ).
Forgot to give you the cross-reference. I talk about the rule for dividing powers here: https://barrygarelick.substack.com/p/ae-11-traditional-math-division-and?utm_source=url
Actually, the priors have been introduced in an earlier chapter, so I should mention that. I talk about the rule of subtracting exponents when dividing earlier. So what you describe has already been presented.
This is all good. I think my own approach would differ slightly, but it might require more work later -- at least in principle -- when one moves on to non-integer exponents. I'd probably start with establishing the obvious extension of equivalent fractions from positive integer numerator/denominator to arbitrary or unspecified values x,a,b: (xa)/(xb) = a/b (x≠0). Then I'd do powers a few like x^2/x^1 = (xx)/(1x) = x/1 = x and x^2/x^3 = (xx)/(xxx) = (x)/(xx) = 1/x = x^(-1) etc and eventually extend to a rule via iteration that x^a/x^b = x^(a-b) when a≥b and 1/x^(b-a) when a≤b, then observe that both may be expressed in the form x^(a-b) when allowing negative exponents. As always what one can (or maybe one should say "should") or cannot do really depends on priors: what have students mastered before the lesson/unit begins? I like the principle that a mathematical problem I solve in the present becomes a tool I use in the future, which is, I know, anathema to the progressive framework of math ed (hey but that's *only one* of its merits :-) ).