Powers of Products and Quotients
Lesson 5 of 16
Objective: SWBAT simplify Powers of Products and Quotients
I begin class by projecting the Launch Power Slip on the whiteboard. I ask students to read carefully and answer the three questions. I call volunteers to the board to share their work.
Question 1: I plan to tell students they can use repeated multiplication to find the prime factorization. They should be able to write:
(22 · 3 · 53) (22 · 3 · 53) (22 · 3 · 53)
(26 · 32 · 59)
When the students share their answers on the board, I ask them which property of numbers allows them to multiply the powers, here. Some students usually say, "the Commutative Property." Others will not recall, so I will probably need to remind them that the correct term is the Associative Property.
Question 2: I expect students will multiply (22 · 3 · 53) (2 · 34) and conclude (23 35 53)
Question 3: I will remind students that for this question, they can use the quotient of powers property. They should conclude 23 32 53
I open the presentation of new material by projecting the resource New Info 1&2 and asking students to analyze what they see. I will call on different students to explain what happens in each row of the first example. I anticipate that students will answer with something along the lines of:
- expanding or repeated multiplication
- grouping like terms (distributive property)
- writing the factors as single powers
I will ask the class to show me thumbs up, thumbs down, or sideways to gauge understanding of what occurs at each step before proceeding.
As a next step, I ask the students to do the same with (3x)4. I will encourage my students to compare the original expressions in both cases (xy)4 and (3x)4 with the final ones x4y4 and 81x4. I will ask them to make a general statement about what happened in each expression. Students will many times say, “the power is distributed.”
If they do, I will ask them if they can write a general rule for the pattern they saw here, using variables. They should come up with something like (ab)n = anbn. We will stay at this task until we arrive at a correct statement of the Power of a Product Property.
I think that this is a good point in time to remind students that in the order of operations, powers take precedence over multiplication and division. For example, in 3x4, the power takes precedence over the multiplication. However, in (3x)4 the multiplication is inside parentheses, which takes precedence over powers.
If time allows, I like to ask students to prove that (3x)4 = 81x4 by substituting a value for x and following the correct order of operations.
- Students many times forget to apply the exponent to all of the factors in expressions like (3x)4 . The coefficient is the term that is often neglected. It’s a good idea to tell students that this mistake is often made. I find that this encourages students to be more careful when simplifying. When a student makes this mistake, I usually ask him/her to substitute a value for x and check their work. Checking their work enables students to understand that they have produced two non-equivalent reflections.
- I find it is important to also remind students that the Power of a Product Property applies only to products, not to sums. In other words, (ab)n = anbn, but ( a + b )n ≠ an + bn.
I follow a similar flow of activity to introduce students to the Power of Quotient Property. I will use Example 2 from New Info 1&2.docx. Afterward, I will ask to students to simplify (2/3)5
I encourage the students to once again try to figure out the general rule on their own. I expect students will be able to figure this property out. We have been working in this manner for several days. Many will say that you "distribute" the exponent among the values or variables in the numerator and denominator of the fraction. In this case students should conclude with (a/b)m = am/bm
It is very important to stress to students that the property works for all non-zero values of a and b.
For this segment of class I pair up students. I try to place students with partners at more or less the same level of mastery of the properties of exponents. This strategy raises the opportunity for students to contribute equally to the work. Then, I hand each student with an Application Powers of Products Quotients handout.
In this application section, students demonstrate their understanding of the exponent properties in this, and previous lessons. The tasks posed to them are to rewrite expressions in equivalent, simpler forms. I like to encourage students to state the name of the relevant property for each step as they work. The goal at this point is for students to develop fluency with the process of simplification.
To close the lesson, I write the following expression on the board: Closure Expression.docx
I ask students to demonstrate their understanding of the properties of exponents by describing how to rewrite the expression without parentheses and simplify it as much as possible. I encourage students to state the name of the property used for each step. But, learning the name of the property is really not the most important aspect. At this point, the goal is using the property to simplify expressions correctly.
Students may multiply
thinking that the Distributive Property is used here. If they do so, I remind the class that the Distributive Property is used over addition or subtraction, but not multiplication.