Lesson 8
Combining Bases
8.1: Same Exponent, Different Base (5 minutes)
Warm-up
The purpose of this warm-up is to encourage students to relate expressions of the form \(a^n \boldcdot b^n\) to \((a \boldcdot b)^n\) by exploring the structure of the factors (MP7). Students should notice that the factors in the expanded form of \(5^3 \boldcdot 2^3\) can be rearranged and multiplied to show the factors in the expanded form of \(10^3\). Evaluating and expanding expressions with exponents helps prepare students for the next activity in which they more generally explore products of bases with the same exponent.
Launch
Give students 2 minutes of quiet work time followed by a whole-class discussion.
Student Facing
- Evaluate \(5^3 \boldcdot 2^3\)
- Evaluate \(10^3\)
Student Response
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Activity Synthesis
Consider asking some of the following questions to focus the conversation on the common exponents:
- “What connections do you see between the two expressions?” (The product of the bases in the first expression is equal to the base in the second expression: \(2 \boldcdot 5 = 10\). The exponents are the same in both expressions.)
- “Is there a way to tell just by looking at the expressions that they would be equal? How?” (Since there are 3 factors that are 5 and 3 factors that are 2, group the 2s and 5s together to get 3 factors that are 10.)
Highlight student explanations that clearly show the connection between \(2^3 \boldcdot 5^3\) and \(10^3\) by inspecting their factors.
8.2: Power of Products (15 minutes)
Activity
Students use repeated reasoning to discover the rule \((a \boldcdot b)^n = a^n \boldcdot b^n\) (MP8).
Launch
Arrange students in groups of 2. Encourage students to share their reasoning with their partner as they work to complete the table. Give students 10–12 minutes of work time followed by a whole-class discussion.
Supports accessibility for: Memory; Organization
Student Facing
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The table contains products of expressions with different bases and the same exponent. Complete the table to see how we can rewrite them. Use the “expanded” column to work out how to combine the factors into a new base.
expression expanded exponent \(5^3 \boldcdot 2^3\) \(\begin{align}(5 \boldcdot 5 \boldcdot 5) \boldcdot (2 \boldcdot 2 \boldcdot 2) &= (5 \boldcdot 2)(5 \boldcdot 2)(5 \boldcdot 2)\\ &= 10 \boldcdot 10 \boldcdot 10 \end{align}\) \(10^3\) \(3^2 \boldcdot 7^2\) \(21^2\) \(2^4 \boldcdot 3^4\) \(15^3\) \(30^4\) \(2^4 \boldcdot x^4\) \(a^n \boldcdot b^n\) \(7^4 \boldcdot 2^4 \boldcdot 5^4\) - Can you write \(2^3 \boldcdot 3^4\) with a single exponent? What happens if neither the exponents nor the bases are the same? Explain or show your reasoning.
Student Response
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Anticipated Misconceptions
Some students may write \(2x^4\) instead of \((2x)^4\). Similarly, students may write \(a \boldcdot b^n\) instead of \((a \boldcdot b)^n\). Ask these students to explain the difference between \(3 \boldcdot 4^2\) and \((3 \boldcdot 4)^2\).
Activity Synthesis
Ask students to share their reasoning about whether \(2^3 \boldcdot 3^4\) can be written with a single exponent. The key takeaway for the discussion should be that the exponents need to be the same to combine the bases into a single base with that exponent.
Introduce a visual display for the rule \((a \boldcdot b)^n = a^n \boldcdot b^n\). Display it for all to see throughout the unit.
Design Principle(s): Optimize output (for generalization)
8.3: How Many Ways Can You Make 3,600? (15 minutes)
Optional activity
At this point, students have worked with many different patterns involving exponents. This activity gives students an opportunity to deepen their thinking by generating different equivalent expressions using the rules of exponents. The process of generating different expressions requires students to understand the numerous ways numbers can be broken into factors and how to combine those factors and express the result using exponents.
Launch
Arrange students in groups of 2–3. Provide students with tools for creating a visual display. There will be several rounds in which students will have to generate multiple expressions equivalent to a specific number. For the first round, students have to generate expressions that are equal to 3,600. Show students how to set up their display with an example, such as the one shown here. As a whole class, come up with an expression equivalent to 3,600 using each of the three rules. If time allows, discuss using prime factorization as a strategy. For an example using the first rule: \(3,\!600 = (600 \boldcdot 6) = (2^3 \boldcdot 3^1 \boldcdot 5^2)(2^1 \boldcdot 3^1) = 2^{3+1} \boldcdot 3^{1+1} \boldcdot 5^2 = 2^4 \boldcdot 3^2 \boldcdot 5^2\). Some simpler examples include:
| \(a^n \boldcdot a^m = a^{n+m}\) | \(\frac{a^n}{a^m} = a^{n-m}\) | \(a^n \boldcdot b^n = (a\boldcdot b)^n\) |
|---|---|---|
|
|
\(\frac{60^5}{60^3} = 60^2\) | \(6^2 \boldcdot 10^2= 60^2\) |
Display these examples for all to see while students are working to generate their own expressions equivalent to 3,600. Set a timer for 1 minute (or other amount, depending on time available) and let students work.
When time is up, pair each group with another group for scoring. It would be beneficial to choose 2 groups to use as examples and demonstrate this process for the whole class:
- A group gets 1 point for every unique expression they found that is equivalent to 3,600. (If the two groups found the same expression, neither group gets a point for it.)
- 2 points for every unique expression that uses negative exponents.
- Students can challenge the other group’s expressions if they think they don’t really equal 3,600, or if the group didn’t use any of the three rules.
In the second round, shift the students’ attention to the number \(\frac{1}{200}\) and have them create another visual display (perhaps on the back of their first visual display). Again, the group gets 1 point for each unique expression equivalent to \(\frac{1}{200}\) except if it uses negative exponents, in which case it gets 2 points.
Play as many rounds of this game as time allows. In subsequent rounds, groups pair up with a different opponent. Consider using the following numbers in different rounds if time permits: 810,000; \(\frac{1}{64}\); 3,375. Leave a few minutes for a brief whole-class discussion.
Supports accessibility for: Memory; Conceptual processing
Design Principle(s): Support sense-making; Maximize meta-awareness.
Student Facing
Your teacher will give your group tools for creating a visual display to play a game. Divide the display into 3 columns, with these headers:
\(a^n \boldcdot a^m = a^{n+m}\)
\(\frac{a^n}{a^m} = a^{n-m}\)
\(a^n \boldcdot b^n = (a \boldcdot b)^n\)
How to play:
When the time starts, you and your group will write as many expressions as you can that equal a specific number using one of the exponent rules on your board. When the time is up, compare your expressions with another group to see how many points you earn.
- Your group gets 1 point for every unique expression you write that is equal to the number and follows the exponent rule you claimed.
- If an expression uses negative exponents, you get 2 points instead of just 1.
- You can challenge the other group’s expression if you think it is not equal to the number or if it does not follow one of the three exponent rules.
Student Response
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Student Facing
Are you ready for more?
You have probably noticed that when you square an odd number, you get another odd number, and when you square an even number, you get another even number. Here is a way to expand the concept of odd and even for the number 3. Every integer is either divisible by 3, one more than a multiple of 3, or one less than a multiple of 3.
- Examples of numbers that are one more than a multiple of 3 are 4, 7, and 25. Give three more examples.
- Examples of numbers that are one less than a multiple of 3 are 2, 5, and 32. Give three more examples.
- Do you think it’s true that when you square a number that is a multiple of 3, your answer will still be a multiple of 3? How about for the other two categories? Try squaring some numbers to check your guesses.
Student Response
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Activity Synthesis
In a whole-class discussion, ask “Explain in your own words: What did you learn about exponents from this activity?”
The main point is that numbers can be broken down into their factors in many ways and the exponent rules can be used to express the same value in many ways.
Lesson Synthesis
Lesson Synthesis
In this lesson, students saw that it is possible to combine bases together as long as the exponent is the same. Students also practiced using all of the rules they know to write many equivalent exponential expressions. The goal of the discussion is mainly to check that students understand why the exponent rule \(a^n \boldcdot b^n = (a \boldcdot b)^n\) works. If there is time and interest, students can also share their observations about what they learned by trying to generate as many equivalent exponential expressions as they can.
Here are questions for discussion:
- “Is it possible to write \(4^5 \boldcdot 5^5\) using a single exponent?” (Yes, \(4^5 \boldcdot 5^5 = 20^5\).)
- “What about \(4^3 \boldcdot 5^5\)?” (No. You could combine 3 factors that are 4 and 3 factors that are 5 to make 3 factors that are 20, but there are still 2 factors that are 5 left over.)
- “When is it possible to combine bases together in a single exponent?” (It is only possible when both bases have the same exponent.)
- “What are some patterns or strategies you saw in the ‘How Many Ways?’ game?”
To close the discussion, it may help to mention that looking for patterns in the factors of numbers has a long tradition in mathematics, with many applications for building better computers and devices. Many ways that computers have been programmed to think are based on patterns of factors. Both hacking and protecting computer networks from hackers are rooted in patterns of factors at a fundamental level. This might be a topic that interested students can further explore.
8.4: Cool-down - Help an Absent Student (5 minutes)
Cool-Down
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Student Lesson Summary
Student Facing
Before this lesson, we made rules for multiplying and dividing expressions with exponents that only work when the expressions have the same base. For example, \(\displaystyle 10^3 \boldcdot 10^2 = 10^5\) or \(\displaystyle 2^6 \div 2^2 = 2^4\)
In this lesson, we studied how to combine expressions with the same exponent, but different bases. For example, we can write \(2^3 \boldcdot 5^3\) as \(2 \boldcdot 2 \boldcdot 2 \boldcdot 5 \boldcdot 5 \boldcdot 5\). Regrouping this as \((2 \boldcdot 5) \boldcdot (2 \boldcdot 5) \boldcdot (2 \boldcdot 5)\) shows that
\(\displaystyle \begin{align}2^3 \boldcdot 5^3 &= (2 \boldcdot 5)^3\\ & = 10^3 \end{align}\)
Notice that the 2 and 5 in the previous example could be replaced with different numbers or even variables. For example, if \(a\) and \(b\) are variables then \(a^3 \boldcdot b^3 = (a \boldcdot b)^3\). More generally, for a positive number \(n\), \(\displaystyle a^n \boldcdot b^n = (a \boldcdot b)^n\) because both sides have exactly \(n\) factors that are \(a\) and \(n\) factors that are \(b\).