In this lesson, students learn how to raise quotients to a power. In presenting the information, I further emphasize how quotients can be broken up into fractions. In so doing, I show the reverse—that multiplying fractions such as x/y by itself is the same as writing (x/y)². This then leads into quotients being raised to powers.
The lesson relies on what they have previously learned about products of powers, and powers of products. A problem like (a²/b⁴)³ is really the powers in the numerator and denominator raised to the third power. They have done this before, but not in a quotient. Although the problems look different than what they’ve seen before, my goal is to make them see that they’ve done this before so there’s nothing really that new here. In fact, Problems 3, 4 and 5 of the Warm-Ups lead.s them to what they will be learning.
Warm-Ups.
Problem 2 requires applying the exponent rule, in which exponents are subtracted. Students will be familiar with Problems 3 and 4 since they have done them before. They act as scaffolding for them to solve Problem 5, which requires raising numerator and denominator to the third power—which they’ve already done in the previous two problems. Prompts may include: “Does the numerator look familiar? What about the denominator? Have you seen a problem similar to it in today’s Warm-Ups?”
Multiplying Fractions. Students remember how to multiply numerical fractions so they will easily answer the question: “How do I multiply 2/3 by 4/5?” Heating 8/15, I write on the board:
“Knowing how to multiply fractions, can you tell me what is the answer to this problem?” (The students generally put it all together, some hesitant, and the rest fairly confident.)
“It’s the same as numerical fractions; multiply across the numerators and then across the denominators. What about if I had something like this?”
“Let’s try one more.”
“If you see something like this, you might want to change x in this case to x/1. Can you multiply it now?” Ultimately, I’d like students to be able to do the multiplication without having to write x as x/1, but students like to hang on to it.
Examples:
Power of a Product Included in Power of Quotients
“Knowing what we know about multiplying x/y by c/d, what would we get if we multiplied x/y by x/y?”
Students should be able to see that the answer is x²/y².
“Can I write x²/y² another way?” If no answer, I provide a hint: x²y²=(xy)² Immediately students will see that the answer is (x/y)².
“You know all this stuff already, because we’re just applying what you know about powers of products to powers of quotients. What’s another way of writing (x/y)³?”
Answer: x³/y³
“We saw for Problem 5 that the general rule is as follows:
“Just break it down by numerator over denominator. We raise the numerator to the x power, and then the denominator to the x power. What if we have this?”
I have them do it in their notebooks, advising them to raise the numerator to the third power, and then the denominator. The answer is:
I’ll write the numerator and denominator separately as an illustration of how I break it down, and then express it as a fraction, so they get the idea that they’ve done this before.
Examples:
I check notebooks to see if there students have done this problem in different ways. If not I’m prepared to show, but if so, I’ll have two students write their method on the board. One way is to simplify the expression inside the parentheses first resulting in (a²)² which is a⁴. Another is to raise numerator and denominator to the second power to obtain:
Homework: Problems should be a mix of finding quotients that are not raised to powers (such as those in the last lesson), and quotients that are. The mix should also include finding product of powers, and power of products.