Power Rules

A Method of Reducing Algebraic Power Rules to Just Two Major Principles

In a basic algebra class, we teach the "rules of exponents" and the "rules of radicals". Altogether, this usually appears in the form of 14 or so separate symbolic rules (and maybe more if you're careful to point out common simplifying errors to avoid).

I've taught this in a rather dramatically different way for several years, in a manner which collapses all the different rules to just two major principles. This is based on some observations I made which seem pretty trivial in retrospect, but I find them to be useful -- not a panacea, but they get some traction from students, and better convey the deep global applicability of math (not a big list of disjointed rules to memorize).

At this point, I tend not to even see what I'm doing as something unusual, but I showed it to a friend of mine the other week who has a PhD in Molecular Biology, and she exclaimed, "I was never taught it that way!", and seemed quite delighted. (To my knowledge, really, no one's ever taught it this way, since it's a method I developed -- for what that's worth.) I've mentioned it in passing before but I figured I would highlight it clearly here today.

Order of Operations (OOP)

First, I start the course by teaching a proper order-of-operations, and tell students that it's the single most important thing in the class -- the "engine that drives everything we do" -- the only thing that is arbitrarily made up, and from which everything else flows. It's not PEMDAS. They must memorize the following chart:

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

Of course, the details are important: Parentheses means "operations inside parentheses" (and includes other grouping symbols like braces, brackets, fraction bars, radical vinculums, and absolute values). In any phase of calculation, we work left-to-right across the expression (just like we read), calculating any of the given operations as we encounter them. Note that after parentheses, each operation comes paired with its inverse (tied in order of operations). As we do exercises, I'm careful to verbally model the mental process: "Do we have any parentheses? No, so we don't need a written line for that. Do we have any exponents or radicals? Yes, so we'll need a written line for that..." Etc.

Principle #1: Operations on same-base powers shift one place down in OOP.

I now call this the "Fundamental Rule of Exponents".
 
When performing any algebraic operation on powers (like x²), we have a simple mental shortcut, and that shortcut is found in the order of operations picture by shifting one place down. Some examples:

Ex. #1: Simplify (x³)². Think: I have a power (x³) and I'm exponentiating it (raising it to the power 2). What is my shortcut? Find "exponent" in the order-of-operations and shift one place down and you see our shortcut: multiply the powers. So (x³)² = x⁶.

Proof: (x³)² = (x∙x∙x)(x∙x∙x) = x⁶ [finding total x's multiplied]

Ex. #2: Simplify x⁵/x³. Think: I have same-base powers (both base x) and they are being divided. What is my shortcut? Find "divide" in the order-of-operations and shift one place down: we will subtract our powers. So x⁵/x³ = x².

Proof: x⁵/x³ = (x∙x∙x∙x∙x)/(x∙x∙x) = x² [cancelling x's top & bottom]

Stuff like that. You can do more initial examples based on student inquiry or interest; of course, it also works for multiplying and radicalizing same-base powers (shortcuts to add and divide, respectively). Maybe weaker students wind up having to memorize all four implied relationships anyway, but I think that's okay. It gives everyone a framework for truly understanding the relationships between operations when they need it.
 
(And there's even something else that I often add later in the course: What happens if we add or subtract powers? Look below "add" and what you see is -- nothing. I'll literally write "No Operation" on a 5th line at this point, and even mention the analogous CPU machine language command. So it's consistent that you'll never be changing powers in an add or subtract operation; simply combine like terms and transcribe the powers.)

Principle #2: Operations distribute over any operation one line down in OOP.

I now call this the "General Distribution Rule".

So what I mean here is that (as I explain in the lecture) you've got parentheses, with one operation outside, and another operation inside. The observation is that we have a very nice, one-line shortcut to get rid of the parentheses by applying the outside operation to each piece inside -- so long as the inner operation is one of the items one line down in the order-of-operations.

Ex. #1: Simplify 7(x+5). Think: We have parentheses. The outside operation is multiplication. The inside operation is addition. Since the latter is one line down in OOP, we can distribute this: namely, "distribution of multiplication over addition". (Note that the "over" in the official name echoes and recalls the relationship in the OOP picture.) So 7(x+5) = 7x+35.

Check: Ask students for a specific value for x, substitute into both sides, and check to see if they are the same value. (Optional: Most students are already comfortable with this distribution, and don't need time spent on the check.)

Ex. #2: Simplify (a²b³)². Think: Once more, we have parentheses. The outside operation is exponentiation. The inside operation is multiplication. Again, since the latter is one line further down in OOP, we can distribute this in a one-line shortcut -- "distribution of exponents over multiplication". So, recalling the first principle for applying exponents to powers: (a²b³)² = a⁴b⁶.

Proof: (a²b³)² = (a∙a∙b∙b∙b)(a∙a∙b∙b∙b) = a⁴b⁶ [total a's and b's multiplied]

Ex. #3: True or False: (x+4)² = x² + 16. Think: Look at the parentheses on the left. The outside operation is an exponent, while the inside operation is addition. This will not distribute in a one-line shortcut, since addition is two lines below exponents. Therefore the statement is False. (Note that this is one of the most common errors in basic algebra, and so it deserves special attention -- I'll write "exponents do not distribute over add/subtract" on the board.)

Check: Ask students for a specific value for x (not zero), substitute into both sides, and check to see if they are the same value.

Of course, this principle also works to recall any of: distributing multiplication over add/subtract, distributing division over add/subtract, distributing exponents over multiply/divide, and distributing radicals over multiple/divide. There's actually a total of a full dozen (12) relationships explained by this one single principle (including the fact that exponents/radicals do not distribute over add/subtract).

Conclusion

It pays off to do as many simplifying examples as time permits afterward, but I do think that getting these two major principles on the board as soon as possible "primes the pump" for everything that happens later. It provides an overarching organizational structure, and it also serves as a model for abstract, generalized principles being easier to remember and apply (making us more happy) than a large body of otherwise disjointed rules. Which from my perspective, may be the only thing that justifies everyone taking a basic algebra course in the first place.