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.

(1) Exponents will multiply powers, i.e., (am)n = am∙n. Example: (x6)2 = x12.
(2) Radicals will divide powers, i.e., n√am = am/n. Example: 3√x15 = x5.
(3) Multiplying will add powers, i.e., am∙an = am+n. Example: x4∙x7 = x11.
(4) Division will subtract powers, i.e., am/an = am−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!


  1. I've seen this presented as the "Exponential Law" or the "Law of Exponents".

    For some stupid reason, there is no standard name for this process.

    1. Interesting, but I don't see either of those with any hits on Google. What I do see a lot of (and common in books) is "Laws of Exponents" in the plural, listing a half-dozen disconnected relationships. It's weird, but I've never seen any place abstract it out to one single concept before.

  2. Just discovered the only other place I've seen online where this same observation is described (without any given name), at the site OakRoadSystems. Nice presentation.