Fundamental Rule of Exponents

For a basic algebra class, given the rudimentary order-of-operations that looks like this:

1. Parentheses
2. Exponents & Radicals
3. Multiplication & Division
4. Addition & Subtraction

 Then we have:

The Fundamental Rule of Exponents: Operations on same-base powers shift one place down in the order of operations.

Cases:
(1) Exponentiation effectively multiplies powers, i.e., (a^m)ⁿ = a^m∙n. Example: (x⁶)² = x¹².
(2) Radicals effectively divide powers, i.e., ⁿ√a^m = a^m/n. Example: ³√x¹⁵ = x⁵.
(3) Multiplication effectively adds powers, i.e., a^m∙aⁿ = a^m+n. Example: x⁴∙x⁷ = x¹¹.
(4) Division effectively subtracts powers, i.e., a^m/aⁿ = a^m−n. Example: x⁹/x² = x⁷.

We've discussed this before, but I just recently decided to apply the name shown here to the pattern. It doesn't show up on a Google search yet, so I think it's fair-game to do so. Cheers!