Lesson 7
Practice with Rational Bases
7.1: Which One Doesn’t Belong: Exponents (5 minutes)
Warmup
This warmup prompts students to compare four exponential expressions. It encourages students to explain their reasoning, hold mathematical conversations, and gives you the opportunity to hear how they use terminology and talk about characteristics of the expressions in comparison to one another. To allow all students to access the activity, each expression has one obvious reason it does not belong. Encourage students to move past the obvious reasons and find reasons based on mathematical properties. During the discussion, listen for important ideas and terminology that will be helpful in the upcoming work of the unit.
Launch
Arrange students in groups of 2–4. Display the expression for all to see. Ask students to indicate when they have noticed one expression that doesn’t belong and can explain why. Give students 1 minute of quiet think time and then time to share their thinking with their small group. In their small groups, tell each student to share their reasoning why a particular expression doesn’t belong and together find at least one reason each expression doesn't belong.
Student Facing
Which expression doesn’t belong?
\(\frac{2^{8}}{2^5}\)
\(\left(4^{\text5}\right)^{8}\)
\(\left( \frac34 \right)^{\text5} \boldcdot \left( \frac34 \right)^{8}\)
\(\frac{10^{8}}{5^5}\)
Student Response
For access, consult one of our IM Certified Partners.
Anticipated Misconceptions
In the last question, students may think \(\frac{10^{8}}{5^5}\) is equal to \(10^3\) or \(\left(\frac{10}{5}\right)^3\). Encourage these students to think about expanding the exponents into their repeated factors.
Activity Synthesis
Ask each group to share one reason why a particular expression does not belong. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question of which one does not belong, attend to students’ explanations and ensure the reasons given are correct.
During the discussion, ask students to explain the meaning of any terminology they use, such as “base” or “exponent.” Also, press students on unsubstantiated claims. For example: “\(\left(4^{\text5}\right)^{8}\) is a really large number.” (It isn’t, because of the negative exponent.) Or, “\(\frac{2^{8}}{2^5}\) and \(\frac{10^{8}}{5^5}\) are whole numbers.” (They are, but how do we know?)
7.2: Exponent Rule Practice (15 minutes)
Activity
This activity develops procedural fluency with exponent rules and encourages students to think about their own learning. Students choose 6 of 12 possible problems to solve, thereby identifying problems that they consider more difficult versus less difficult. Notice which problems students choose more than others, and which problems are skipped more than others. The first set of problems checks whether students can apply the exponent rules procedurally. The next set of problems checks whether students understand what negative exponents mean. The last set of problems asks students to evaluate exponents to check whether they understand the meaning of the zero exponent and the definition of exponents as repeated multiplication (by the base, or by the reciprocal of the base in the case of negative exponents).
Launch
Arrange students in groups of 2. Encourage students to work together with their partners. Encourage partners to choose mostly the same problems, but if they differ, partners should check one another’s work. Encourage students to explain their reasoning by referencing the visual displays for the exponent rules. Give students 10–12 minutes to work followed by a brief wholeclass discussion. Problems that many students chose to skip can be assigned as additional practice.
Supports accessibility for: Memory; Language
Design Principle(s): Optimize output for (explanation)
Student Facing
 Choose 6 of the equations to write using a single exponent:
 \(7^5 \boldcdot 7^6\)
 \(3^{\text3} \boldcdot 3^8\)
 \(2^{\text4} \boldcdot 2^{\text3}\)
 \(\left(\frac{5}{6}\right)^4 \left(\frac{5}{6}\right)^5\)
 \(\frac{3^5}{3^{28}}\)
 \(\frac{2^{\text5}}{2^4}\)
 \(\frac{6^5}{6^{\text8}}\)
 \(\frac{10^{\text12}}{10^{\text20}}\)
 \(\left(7^2\right)^3\)
 \(\left(4^3\right)^{\text3}\)
 \(\left(2^{\text8}\right)^{\text4}\)
 \(\left(6^{\text3}\right)^5\)
 Which problems did you want to skip in the previous question? Explain your thinking.
 Choose 3 of the following to write using a single, positive exponent:
 \(2^{\text7}\)
 \(3^{\text23}\)
 \(11^{\text8}\)
 \(4^{\text9}\)
 \(2^{\text32}\)
 \(8^{\text3}\)
 Choose 3 of the following to evaluate:
 \(\frac{10^5}{10^5}\)
 \(\left(\frac{2}{3}\right)^3\)
 \(2^8 \boldcdot 2^{\text8}\)
 \(\left(\frac{5}{4}\right)^2\)
 \(\left(3^4\right)^0\)
 \(\left(\frac{7}{2}\right)^2\)
Student Response
For access, consult one of our IM Certified Partners.
Activity Synthesis
The goal of the discussion is to get a general sense of how fluent students have become with the exponent rules. Poll the class about how successful they felt while working on the problems. Here are some questions for discussion:
 “Which problems would you assign to your best friend? Why?”
 “Which problems really made you think? Why?”
 “What are some resources you could use to get more comfortable with the problems you are uncomfortable with?”
7.3: Inconsistent Bases (15 minutes)
Activity
In this activity, students analyze powers that involve different bases. The goal is for them to recognize that exponents can be added (or subtracted) only when the powers being multiplied (or divided) have the same base. It is expected that students compute the value of the expressions on the left and right sides of the equation to show they are not actually equal. The last problem alludes to the rule \(a^n \boldcdot b^n = (a \boldcdot b)^n\) which will be explored further in the next lesson.
As students work, notice those who check for equality by computing the value on either side of each equation or by expanding each power into its factors. Invite them to share later.
Launch
Display the false equation \(2^3 \boldcdot 5^2 = 10^{3+2} = 10^5\) for all to see. Ask students whether they think the equation is true or false, and choose a few students to explain their reasoning. If not mentioned by students, expand \(2^3 \boldcdot 5^2\) and \(10^5\) to show their repeated factors. Give students 10 minutes to work followed by a wholeclass discussion.
Supports accessibility for: Memory; Conceptual processing
Student Facing
Mark each equation as true or false. What could you change about the false equations to make them true?

\(\left(\frac{1}{3}\right)^2 \boldcdot \left(\frac{1}{3}\right)^4 = \left(\frac{1}{3}\right)^6\)

\(3^2 \boldcdot 5^3 = 15^5\)

\(5^4 + 5^5 = 5^9\)

\(\left(\frac{1}{2}\right)^4 \boldcdot 10^3 = 5^7\)
 \(3^2 \boldcdot 5^2 = 15^2\)
Student Response
For access, consult one of our IM Certified Partners.
Student Facing
Are you ready for more?
Solve this equation: \(3^{x5} = 9^{x+4}\). Explain or show your reasoning.
Student Response
For access, consult one of our IM Certified Partners.
Activity Synthesis
The important takeaway from this activity is that the exponent rules work because they capture patterns of repeated multiplication of a single base. The equation in the launch erroneously applies an exponent rule to a situation that involves multiple bases. This fails because with multiple bases, there are not the same patterns of repeated multiplication. Ask students to share their responses and display them for all to see. For the final question, ask students whether they think it is a coincidence that the equation is true, or if there is another, more general explanation. It is not necessary to dwell on this point since it will be addressed more fully in the next lesson. Consider involving more students in a wholeclass discussion with the following questions:
 “Who can restate __’s reasoning in a different way?”
 “Why do the exponent rules we have looked at so far only work when looking at one particular base rather than mixing different bases together?” (\(3^2 \boldcdot 3^3 = 3^5\) because there are 5 factors that are 3 on the left side, but \(3^2 \boldcdot 4^3\) isn't \(12^5\) because there are not 5 factors that are 12.)
Design Principle(s): Maximize metaawareness, Optimize output (for explanation)
Lesson Synthesis
Lesson Synthesis
In this lesson, students honed their skills working with exponent rules and discovered where the rules break down when looking at expressions with mismatching bases. Here are some questions for discussion:
 “Why is the equation \(2^5 \boldcdot 2^3 = 2^{15}\) false?” (Multiplying 5 factors that are 2 by 3 factors that are 2 results in a total of 8 factors that are 2. Multiplying the exponents doesn’t make sense in this case.)
 “Why is the equation \(\frac{3^5}{3^2} = 3^3\) true?” (Expanding the left side, we get \(\frac{3 \boldcdot 3 \boldcdot 3 \boldcdot 3 \boldcdot 3}{3 \boldcdot 3}\), which is equal to \(1 \boldcdot 3 \boldcdot 3 \boldcdot 3\) or just \(3^3\).)
 “Why is the equation \(\frac{6^5}{3^2} = 2^3\) false? Why might someone make this mistake?” (Expanding the left side, we get \(\frac{6 \boldcdot 6 \boldcdot 6 \boldcdot 6 \boldcdot 6}{3 \boldcdot 3}\), which is equal to \(2 \boldcdot 2 \boldcdot 6 \boldcdot 6 \boldcdot 6\). Someone might make this mistake because they divide 6 by 3 and use the exponent rule for division to subtract the exponent in the denominator from the exponent in the numerator.)
7.4: Cooldown  Working with Exponents (5 minutes)
CoolDown
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Student Lesson Summary
Student Facing
In the past few lessons, we found rules to more easily keep track of repeated factors when using exponents. We also extended these rules to make sense of negative exponents as repeated factors of the reciprocal of the base, as well as defining a number to the power of 0 to have a value of 1. These rules can be written symbolically as:
\(\displaystyle x^n \boldcdot x^m = x^{n+m},\) \(\displaystyle \left(x^n\right)^m = x^{n \boldcdot m},\) \(\displaystyle \frac{x^n}{x^m} = x^{nm},\) \(\displaystyle x^{\textn} = \frac{1}{x^n},\) and \(\displaystyle x^0 = 1,\)
where the base \(x\) can be any positive number. In this lesson, we practiced using these exponent rules for different bases and exponents.