The previous lesson leads directly into the topic of taking the power of a power, and the power of products. Specifically, students have learned that (a²)(a²)(a²) equals a⁶ because the exponents are added. It is then easy that for (a²)³ the exponents are multiplied. This is then extended to terms like (a²b³)² so that each of the exponents in the parentheses are multiplied by 2.
Students liken this to “distributing the exponent” and textbooks warn teachers to tell students not to characterize it in this manner. The reason is that by avoiding students using this description, it will help prevent them from assuming that they can apply the power to a power rule to binomials raised to a power; e.g.:
(I put the “Not true!!” warning above in case someone quickly looked at this introduction and assumed I had made a mistake.)
It has been my experience that there are students who will make the above mistake no matter how many warnings are given. Telling students that the power of a product like (a²b³)² is not “distributing the 2 to the other exponents” is not likely to stop them from making the mistake with binomials.
A friend who teaches high school math tells me she uses the word “apply” instead of distribute: “The exponent is applied to the numbers and variables inside the parentheses which is a good way to refer to it. I’ve found, however, that the distribution analogy for power of a product tends to help them see how the procedure works.
That said, I do issue a warning that this “distribution analogy” does not hold for binomials, so don’t do it. Even so, students look at it as a distribution, so I try to take advantage of the analogy while trying to avoid the mathematical canoe I’m paddling, and the students being carried in it, from smashing into the rocks.
Warm-Ups.
Problems 3, 4 and 5 will be used in the introduction to the day’s lesson. Students are to apply the rules for multiplying powers that they learned previously. For Problem 3, the rule would specify that the product of power of the same base is the base raised to the sum of the exponents. Problem 5 is included to remind students that a variable that does not have an exponent has an exponent of 1 that is not written. Therefore h can be expressed as h¹.
Power of a Power. Having gone over the Warm-Ups, I use them now to talk about the power of a power but first I frame it in terms of exponential form.2 “What is the shorter way to write x∙x∙x? That is, how can I write it as a power?”
They know by now that it is x³.
“Looking at Problem 3 of the Warm-Ups, how can I write (x³)(x³)(x³) as a power?”
The answer comes hesitantly; that is, the answer is stated in the tone of a question because they haven’t seen it in this form before: (x³)³.
“Absolutely correct,” I say. “It may look a bit strange, but it makes sense that we can write it that way because (x³) is used as a factor three times, so it can be written as being raised to the third power. And we saw in the Warm-Up that we added the exponent 3 times. But we can also write this as a product: 3 times 3 can we note?”
There is general agreement.
“What if I had written (x³)⁴ ? What should I do?”
Most see the pattern and say the exponents should be multiplied.
I now write the rule of exponents for a power of a power:
I have them look at Problem 4 of the Warm-Ups. “You can see that we have the same thing going on here. How could we write (2a²b) (2a²b) in exponential form?”
They will answer somewhat hesitantly but a little more confident than before: (2a²b)²
“And we see that each number and variable inside the parentheses is raised to the second power. Remember that 2 can be written as 2¹, and so can b. So when we apply the rule for power of a power we get this:”
“Each number of variable inside the parentheses is raised to the power of 2. Which means that we multiply exponents as I’ve shown. Let’s try one together:”
“We could write this as 81x⁴. What about this?”
“We are raising -3 to the fourth power. What if I had written it this way?”
“We would get -81x⁴. Why?”
I hear things like “The negative sign is outside the parentheses” or “We’re not raising -3 to the fourth power” or “The negative sign means the whole thing is negative.” These are fairly good responses.
More Examples.
Homework. I work with them on the homework problems which should lead with problems of powers of products, and then a mix of problems from yesterday’s and today’s lesson. Students will tend to get confused, so as necessary I will sometimes devote an extra day to working these type of problems.