The Product of Powers Property
Lesson 1 of 16
Objective: SWBAT use and apply the Product of Powers property of exponents.
Before students enter the classroom, project the Product of Powers Launch Slip slip on the board and ask students to follow the steps provided. Allow them to work with their elbow partner.
In step one, students should list their answers
2; 4; 8; 16; 32; 64; 128; 256; 512; 1,024
Students will tussle with Step Two. The idea of why multiplying any of the first five numbers results in a number also in the list will not be obvious at first. Allow the class to share their ideas with each other, not out loud to the whole class. The answer you are looking for is something like “multiplying the powers will give you a product that is also a result of repeatedly using 2 as a factor”. Students will express this in their own language. You may or may not want to take this as an opportunity to work on precision of language.
Next, ask….”Is this only true for the first five powers of 2? Can we include the 6th power as well?” Allow them to try it. Then, call on a couple of students to share their thinking.
In Step 3, students should be curious to see if the same pattern holds true. Encourage their curiosity!
Once the launch is complete, project the EXPONENTS NEW INFO SLIP. Ask the students to turn and work with their other elbow partner. Allow the pairs about 5-10 minutes to answer the 3 problems.
As the students work they should write,
z3 ∙ z4
z∙z∙z ∙ z∙z∙z∙z (7 z’s) so, z7
After students complete all three tasks, call students to come to the board and share their answers. For Part 2, someone should state that the exponent in the answer is the sum of the exponents in each factor. Ask him or her to try to write a general rule on the board for this pattern (using variables only). When the correct rule is written, state that this is the Product of Powers Property of Exponents.
Two common mistakes are made with Part C. One is that students multiply the bases and add the exponents, writing 256. The second mistake is to not follow the instructions explicitly (writing the the numerical answer 15625). MP6 focuses on attention to precision. An important aspect of mathematical precision is reporting results in the required/desired format.
Before moving on ask the class if they can relate what they did here to the launch questions projected at the start of the class. Ask another student to explain Step 2, using an example and the newly named Product of Powers property. Students should again state that the product of any two powers is also on the list and provide an example such as:
4 ∙ 16 = 64
22 ∙ 24 = 26
Demonstrating that 22 ∙ 24 = 22 + 4
Handout an Applying the Property Handout sheet to each student. There are 7 questions which students must answer. The problems encourage them to apply the Product of Powers Property.
Watch for the following common mistakes:
- Students may multiply the bases in question 2, [(−7)3 · (−7)2] and write 495. If they do, ask them to expand each power (−7)(−7(−7) · (−7(−7)
- Students may add 3 and 5 in question 6 [3a4 · 5a2] instead of multiplying these coefficients. If they do, remind students that the 3 and the 5 are coefficients of the variable and therefore they are factors in the overall expression that must be multiplied.
- Students may take 3 to the fourth power and 5 to the second power. Remind students that the coefficient in this case is not part of the base, and the only numbers that should be added are the exponents of a.
- You may want to expand the expression 3 · a4 · 5 · a2 and indicate that the Commutative Property allows you to rewrite it as 3 · 5 · a4 · a2
Question 7 is tricky. Observe students and remind them that the Product of Powers Property applies only when the bases are the same. Substituting the variable bases for numbers may be a good way to demonstrate this on the board.
To end the lesson, quickly write the following instruction on the board.
Write as many multiplication statements you can that use the Product of Powers property to get an answer of 108
Allow a couple of minutes and then ask students to call out their answers as you point to them.
1. Ask students to demonstrate that b0 = 1 using the Power of Products property.
If necessary, watching the video below may help.
2. This video uses a pattern to demonstrate why any base to the zero power equals 1.
Students should be asked to demonstrate the same pattern with bases other than 3.
and then asked to return to question 1.
3. Here are extra practice problems involving the zero exponent property.
1. x3 ∙ x0 ∙ x7
2. 5n2 ∙ 6n3 ∙ 2n0
3. a5 ∙ b0 ∙ a2 ∙ 2b2