P8 Traditional Math: Zero and Negative Exponents

Eighth Grade

When teaching students about simplifying expressions such as b²/b⁴ by “subtracting up” (i.e., 1/b⁽⁴ ⁻ ²⁾ or 1/b²) I inevitably get asked “Isn’t it also “b⁻ ²?”  I answer that it is, but for now we’ll do it this way, and later we will learn negative exponents. I had a student who was very stubborn about it and asked “What if we write expressions with negative exponents?”  I replied that until we get to the lesson on negative exponents, I would take off points on a quiz if negative exponents were used, and that was the end of the discussion.

I use the rules of exponents (presented in an earlier lesson) to show what negative exponents represent, as well as what the zero power represents. I have seen approaches in which both the zero power and negative exponents are explained in terms of powers of ten. Specifically, 10³ is 1/10 of 10⁴, 10² is 1/10 of 10³ and so on. When we get to 10¹ (or  10), then the pattern tells us that 10⁰ must be 1/10 of 10 or 1. When we go further to the right on this powers-of-ten number line, then 10⁻¹ is 1/10 of 1, and so on. I’ve found that using the exponent rules to explain zero and negative exponents are clearer to students.

Another approach I’ve seen for the zero power is showing that x³, say, is a number in which x is a factor three times—but multiplied by 1: that is, 1∙3∙3∙3.  Then 3¹ is 1∙3, and 3⁰ = 1, since 3 is used as a factor zero times. I will sometimes refer to this explanation when students try to articulate what a zero power represents.

Warm-Ups.

These problems go over what was learned primarily in the last lesson. Problem 4 reinforces the idea that 1 divided by any number is the reciprocal of that number. Thus, the reciprocal of a/b is 1/(a/b) which is b/a.

Zero Exponents.  “What do you think 5⁰ is?” I will ask my students who almost always respond “Zero!” and to which I reply “Let’s look at this more closely.”

I write on the board:

After getting agreement that any number divided by itself is 1, I then ask how we would use the rule of exponents for quotients, reminding them that the rule calls for subtracting exponents.  They quickly see that it is equal to 5⁰.

“Now if 5⁰ equals 5⁴/5⁴, and 5⁴/5⁴ equals 1, if I apply the rule of transitivity, what can we say about 5⁰?”

The light goes on at this point, although I’ve had some students ask what the rule of transitivity is. I have to remind them. (Forgetting is a common part of math classes, which is why we have to keep repeating things they’ve learned.)

I write the following:

“You may be wondering why I say that b cannot equal zero.  Let’s take a look. If I have zero to any power, the answer is zero. So let’s see what happens if I have this:

“We all agree that this is 0/0, but if you recall from last year, what did we say about dividing zero by zero?”

Some may remember, but someone usually will say “It equals 1.”  And I will respond, "Can it equal 2?  How about 5?  What is zero times any number?” 

Zero divided by zero is “indeterminate” since there is no single number that satisfies it—thus we cannot have zero to the zero power.

Examples: These serve to expand how the zero power is used in expressions.

Negative Exponents.  “Somebody simplify these for me; write it on your mini-whiteboards.”

They’ve been doing problems like this the past few days so they come up with the answers fairly quickly: a⁴ and 1/a⁴.

“Some of you asked if we could subtract exponents and express the answer as a⁻⁴. And as a matter of fact, you can.  If you recall, Problem 4 of the warm ups had 1/(a/b). What did we call it when we divide 1 by any number? The result is called what?”

Hearing “reciprocal”, (with a number of students again forgetting what was just talked about recently) I continue on undaunted.  “So what’s the reciprocal of a³/a⁷?”

Some will respond a⁴ and others will respond 1/(a³/a⁷) and still others a⁷/a³.  “Good; all are correct so we’ll work with all of them.” I write on the board:

I walk them through the above starting with the pair on the left, asking what a³/a⁷ is writing it, and then I complete the equation, stating that 1/(a⁷/ a³) is 1/a⁴. I sometimes explain it as “a⁻⁴ has a negative attitude, so we’ll send him downstairs to think about his behavior. When we do, we get 1/a⁴ so his attitude has improved.”

I now write:

“If I said I want to write 1/a⁻⁴ so that I have no negative exponents, I would write a⁴. They are the same thing.  How would I write 1/x⁻³ with a positive exponent?” 

The answer of course is x³.

Common Mistake. Students sometimes will assume that the negative exponent turns the expression into a negative one. I emphasize that this is not true.

Examples.

This one I walk through with the students. This lends itself to a short-cut: a fraction raised to a negative power is the reciprocal of that fraction raised to a positive power. But I do not introduce that yet, since they are processing a lot of information now. Later I will talk about it and have more complex examples.

Homework.  The homework is a mix of zero and negative exponents, keeping them fairly straightforward as in the examples. More complex problems are introduced at the end as challenge problems, such as: Express with positive exponents: