Lesson 6
What about Other Bases?
6.1: True or False: Comparing Expressions with Exponents (5 minutes)
Warmup
The purpose of this warmup is for students to reason about powers of positive and negative numbers. While some students may find the numeric value of each expression to make a comparison, it is not always necessary. Encourage students to reason about the meaning of the integers, bases, and exponents rather than finding the numeric value of each expression.
Launch
Display one problem at a time. Give students up to 1 minute of quiet think time per problem and tell them to give a signal when they have an answer and a strategy. Follow with a wholeclass discussion.
There may be some confusion about how negative numbers interact with exponents. Tell students that exponents work the same way for both positive and negative bases. In other words, \((\text 3)^2 = \text 3 \boldcdot \text 3\).
Students may also struggle to interpret the notation difference between two expressions such as \((\text 4)^2\) and \(\text 4^2\). Tell students that the interpretation of the first expression is \((\text 4)^2 = \text 4 \boldcdot \text 4\) and the interpretation of the second expression is \(\text 4^2 = \text 1 \boldcdot 4 \boldcdot 4\).
Student Facing
Is each statement true or false? Be prepared to explain your reasoning.

\(3^5 < 4^6\)

\(\left(\text 3\right)^2 < 3^2\)

\(\left(\text 3\right)^3 = 3^3\)

\(\left( \text 5 \right) ^2 > \text 5^2\)
Student Response
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Activity Synthesis
Poll students on their response for each problem. Record and display their responses for all to see. If all students agree, ask 1 or 2 students to share their reasoning. If there is disagreement, ask students to share their reasoning until an agreement is reached. After the final problem, ask students if the same strategies would have worked if all of the bases were 10 to highlight the idea that powers with a base of 10 behave the same as those with bases other than 10.
To involve more students in the conversation, consider asking:

“Do you agree or disagree? Why?”

“Who can restate ___’s reasoning in a different way?”

“Does anyone want to add on to _____’s reasoning?”
6.2: What Happens with Zero and Negative Exponents? (15 minutes)
Activity
Extend the definition of nonpositive exponents to other bases. The work of this activity is parallel to the work with the powers of 10 table in the previous lesson. Notice students who transfer their understanding of powers of 10, especially nonpositive exponents, to exponential expressions with bases other than 10.
Launch
Arrange students in groups of 2. Tell students to work on their own and then share their reasoning with their partner after each has had a chance to complete the problem. Encourage students to look for similarities to work they have already done with base 10. Give students 10 minutes of work time, followed by a whole class discussion.
Supports accessibility for: Visualspatial processing; Organization
Design Principle(s): Support sensemaking; Maximize metaawareness
Student Facing
Complete the table to show what it means to have an exponent of zero or a negative exponent.
 As you move toward the left, each number is being multiplied by 2. What is the multiplier as you move toward the right?
 Use the patterns you found in the table to write \(2^{\text 6}\) as a fraction.
 Write \(\frac{1}{32}\) as a power of 2 with a single exponent.
 What is the value of \(2^0\)?
 From the work you have done with negative exponents, how would you write \(5^{\text 3}\) as a fraction?
 How would you write \(3^{\text 4}\) as a fraction?
Student Response
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Student Facing
Are you ready for more?

Find an expression equivalent to \(\left(\frac{2}{3}\right)^{\text3}\) but with positive exponents.

Find an expression equivalent to \(\left(\frac{4}{5}\right)^{\text8}\) but with positive exponents.

What patterns do you notice when you start with a fraction raised to a negative exponent and rewrite it using a single positive exponent? Show or explain your reasoning.
Student Response
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Activity Synthesis
The important idea for this activity is that the same logic that we used to analyze powers of 10 applies to other bases as well. Students will work mostly with bases that are whole numbers and positive fractions.
Here are some sample questions for wholeclass discussion:
 “How does working with powers of 2 compare to working with powers of 10?” (The properties apply in the same way, but computing the value of the exponents is more difficult.)
 “What happens when you use exponent rules when one of the exponents is 0?” (The rules apply in the same way. \(2^0 = 1\) in the same way that \(10^0 = 1\).)
 “How does \(10^{\text 3}\) compare to \(2^{\text 3}\)?” (Both values are positive and less than 1. Since \(10^{\text 3}\) has greater numbers in the denominator, it is less than \(2^{\text 3}\).)
Introduce new (or update current) visual displays to illustrate exponent rules for positive, rational bases. Display them for all to see throughout the unit. An example would be to update the rule \(10^n \boldcdot 10^m = 10^{n+m}\) with \(x^n \boldcdot x^m = x^{n+m}\) with a guiding example that uses a base other than 10.
6.3: Exponent Rules with Bases Other than 10 (15 minutes)
Activity
This activity gives students a chance to practice using exponent rules to analyze expressions and identify equivalent ones. Five lists of expressions are given. Students choose two lists to analyze. As students work, notice the different strategies used to analyze the expressions in each list. Ask students using contrasting strategies to share later.
Launch
Arrange students in groups of 2. Tell students to take turns selecting an expression, determining whether the expression matches the original, and explaining their reasoning. Give students 10 minutes to work followed by a wholeclass discussion.
Student Facing
Lin, Noah, Diego, and Elena decide to test each other’s knowledge of exponents with bases other than 10. They each chose an expression to start with and then came up with a new list of expressions; some of which are equivalent to the original and some of which are not.
Choose 2 of the 4 lists to analyze. For each list of expressions you choose to analyze, decide which expressions are not equivalent to the original. Be prepared to explain your reasoning.
 Lin’s original expression is \(5^{\text9}\) and her list is:
\((5^3)^{\text3}\)
\(\text5^9\)
\(\frac{5^{\text6}}{5^3}\)
\((5^3)^{\text2}\)
\(\frac{5^{\text4}}{5^{\text5}}\)
\(5^{\text4} \boldcdot 5^{\text5}\)
 Noah’s original expression is \(3^{10}\) and his list is:
\(3^5 \boldcdot 3^2\)
\((3^5)^2\)
\((3 \boldcdot 3)( 3 \boldcdot 3)( 3 \boldcdot 3)( 3 \boldcdot 3)( 3 \boldcdot 3)\)
\(\left(\frac{1}{3}\right)^{\text10}\)
\(3^7 \boldcdot 3^3\)
\(\frac{3^{20}}{3^{10}}\)
\(\frac{3^{20}}{3^2}\)
 Diego’s original expression is \(x^4\) and his list is:
\(\frac{x^8}{x^4}\)
\(x \boldcdot x \boldcdot x \boldcdot x\)
\(\frac{x^{\text4}}{x^{\text8}}\)
\(\frac{x^{\text4}}{x^8}\)
\((x^2)^2\)
\(4 \boldcdot x\)
\(x \boldcdot x^3\)
 Elena’s original expression is \(8^0\) and her list is:
1
0
\(8^3 \boldcdot 8^{\text3}\)
\(\frac{8^2}{8^2}\)
\(10^0\)
\(11^0\)
Student Response
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Anticipated Misconceptions
Some students may struggle with comparing \(\text5^9\) to \(5^{\text9}\). Ask these students to explain the difference between \(10^3\) and \(10^{\text3}\), and relate that to the difference between \(5^9\) and \(5^{\text9}\). If needed, explain that \(\text5^9\) is a large negative number, whereas \(5^{\text9}\) is a very small positive number.
Activity Synthesis
The key idea is that the exponent rules are valid for nonzero bases other than 10. Select students to share which expressions did not match the original in the lists they chose to study. Consider focusing on questions 3 and 4 to assess how students have generalized in terms of variables and whether students have internalized the 0 exponent. Select previously identified students to share the different strategies used to analyze the same list.
Here are some questions to consider for discussion:
 “What mistakes might lead to the expressions that are not equivalent to the original expression?”
 “How do you know that expression is not equivalent to the original?”
 “Did anyone think about that expression in a different way?”
Design Principle(s): Support sensemaking; Optimize output for (explanation)
Lesson Synthesis
Lesson Synthesis
The purpose of the discussion is to check whether students understand that the rules they have developed for doing arithmetic with powers of 10 are valid for any other nonzero rational bases as well. This is also a good opportunity to address possible misconceptions students have, especially regarding nonpositive exponents.
Consider using these discussion questions to emphasize the important concepts in this lesson:
 “We saw before that \(10^3\) shows the repeated multiplication \(10 \boldcdot 10 \boldcdot 10\) and \(10^{\text 3}\) shows the repeated multiplication \(\frac{1}{10} \boldcdot \frac{1}{10} \boldcdot \frac{1}{10}\). How do \(2^3\) and \(2^{\text 3}\) compare to \(10^3\) and \(10^{\text 3}\)?” (When the base is 2 rather than 10, the repeated multiplication works the same way. \(2^3\) is \(2 \boldcdot 2 \boldcdot 2\) and \(2^{\text 3}\) is \(\frac{1}{2}\boldcdot \frac{1}{2}\boldcdot \frac{1}{2}\).)
 “How do \(5^3\) and \(5^{\text 3}\) compare to \(10^3\) and \(10^{\text 3}\)?” (The exponent rules work in exactly the same way. The base is just 5 instead of 10.)
 “How is \((\text 3)^4\) different than \(\text 3^4\)? If you get stuck, think about what base is being repeatedly multiplied.” (\((\text 3)^4\) is \((\text 3)(\text 3)(\text 3)(\text 3)\) whereas \(\text 3^4\) is \(\text (3 \boldcdot 3 \boldcdot 3 \boldcdot 3)\).)
 “Do the other exponent rules work the same way that they did for base 10? Give some examples with different bases.” (Yes, the other exponents rules work the same way. With base 7 for example, we have that \(\frac{7^5}{7^2} = 7^3\) and \(7^3 \boldcdot 7^6 = 7^9\).)
 “What about combining bases? For example, do you agree that \(3^5 \boldcdot 4^3 = 12^8\)?” (No, you cannot combine bases in this way. To see this, expand the factors of both sides. The left side has 5 factors that are 3 and 3 factors that are 4. One could combine some of these factors to make 12, but the right side has 8 factors that are 12. They are not equal.)
Close by telling students that the exponent rules with other bases work exactly the same way as they did with 10 and they can always check this by expanding the factors of an expression with exponents (unless the exponent is very large and it would take too long to expand).
6.4: Cooldown  Spot the Mistake (5 minutes)
CoolDown
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Student Lesson Summary
Student Facing
Earlier we focused on powers of 10 because 10 plays a special role in the decimal number system. But the exponent rules that we developed for 10 also work for other bases. For example, if \(2^0=1\) and \(2^{\text n} = \frac{1}{2^n}\), then
\(\begin{align}2^m \boldcdot 2^n &= 2^{m+n} \\ \left(2^m\right)^n &= 2^{m \boldcdot n} \\ \frac{2^m}{2^n} &= 2^{m\text n}. \end{align}\)
These rules also work for powers of numbers less than 1. For example, \(\left(\frac{1}{3}\right)^2 = \frac{1}{3} \boldcdot \frac{1}{3}\) and \(\left(\frac{1}{3}\right)^4 = \frac{1}{3} \boldcdot \frac{1}{3} \boldcdot \frac{1}{3} \boldcdot \frac{1}{3}\). We can also check that \(\left(\frac{1}{3}\right)^2 \boldcdot \left(\frac{1}{3}\right)^4 = \left(\frac{1}{3}\right)^{2+4}\).
Using a variable \(x\) helps to see this structure. Since \(x^2 \boldcdot {x^5} = x^7\) (both sides have 7 factors that are \(x\)), if we let \(x = 4\), we can see that \(4^2 \boldcdot 4^5 = 4^7\). Similarly, we could let \(x = \frac{2}{3}\) or \(x = 11\) or any other positive value and show that these relationships still hold.