Lesson 4
Positive Rational Exponents
4.1: Math Talk: Regrouping Fractions (5 minutes)
Warmup
The purpose of this Math Talk is to elicit strategies and understandings students have for decomposing fractions into a unit fraction times a whole number and using the associative and commutative properties. These understandings help students develop fluency and will be helpful later in this lesson when students will need to be able to decompose a rational exponent into a unit fraction times a whole number.
These questions invite students to notice and make use of commutativity and decomposition of rational numbers (MP7).
Launch
Display one problem at a time. Give students quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a wholeclass discussion.
Supports accessibility for: Memory; Organization
Student Facing
Find the value of each expression mentally.
\(\frac12 \boldcdot 5 \boldcdot 4\)
\(\frac52 \boldcdot 4\)
\(\frac23 \boldcdot 7 \boldcdot \frac32\)
\(7 \boldcdot \frac53 \boldcdot \frac37\)
Student Response
For access, consult one of our IM Certified Partners.
Activity Synthesis
Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:
 “Who can restate _____’s reasoning in a different way?”
 “Did anyone have the same strategy but would explain it differently?”
 “Did anyone solve the problem in a different way?”
 “Does anyone want to add on to _____’s strategy?”
 “Do you agree or disagree? Why?”
Design Principle(s): Optimize output (for explanation)
4.2: You Can Use Any Fraction As an Exponent (10 minutes)
Activity
In this activity, students break down fractional exponents into a unit fraction times a whole number and rewrite the expressions using radicals. The first problem suggests how to break down fractions into a whole number times a unit fraction, and students can apply the same reasoning in subsequent questions.
Look for groups that write equivalent radical expressions that are different from each other to compare during the discussion.
Launch
Display the expression \(5^{\frac23}\)for all to see. Tell students that mathematicians have a convention for what this expression means, and invite them to make a conjecture about what it means. Emphasize that there are no wrong answers; we are just curious about what the expression might mean. Write each conjecture that is shared near the expression. Examples of conjectures might include \(\sqrt[3]5^2\), \(\sqrt5^3\), or \(\frac{2}{5^3}\). Tell students that in this activity, they will use the properties of exponents to understand how mathematicians have decided to interpret this expression.
Arrange students in groups of 2. Tell students there are many possible answers for the questions. After quiet work time, ask students to compare their responses to their partner’s and decide if they are both correct, even if they are different. Follow with a wholeclass discussion.
Design Principle(s): Optimize output (for justification); Cultivate conversation
Supports accessibility for: Memory; Conceptual processing
Student Facing
 Use exponent rules to explain why these expressions are equal to each other:
 \(\left(5^{\frac13}\right)^2\)
 \(\left(5^2\right)^{\frac13}\)
 Write \(5^{\frac23}\) using radicals.
 Write \(5^{\frac43}\) using radicals. Show your reasoning using exponent rules.
Student Response
For access, consult one of our IM Certified Partners.
Anticipated Misconceptions
If students have trouble writing the expressions using radicals, help them make connections to their previous understanding of relationships between cubes and cube roots. It may help to start with an example with friendlier numbers, for instance, “If we know that 5^{3} is 125, what does that tell us about the cube root of 125? And how could we write that with radicals?” (That tells us the cube root of 125 is 5, so we can write 5 as \(\sqrt[3]{125}\).) “If we cube \(5^{2/3}\), what does that tell us about cube roots? And how could we write that with radicals?” (We get 5^{2}, which tells us that the cube root of 5^{2} is \(5^{2/3}\). So we can write \(5^{2/3}\) as \(\sqrt[3]{5^2}\).)
Activity Synthesis
Select previously identified groups to share their responses to the last two questions. Make sure to highlight the fact that these expressions can be written differently depending on how students apply the exponent rules. Display \(\sqrt[3]{5^4}\) and \(\left(\sqrt[3]{5}\right)^4\) for all to see. Ask students, “What happens if we cube both numbers?” (Cubing \(\sqrt[3]{5^4}\) gives \(\left(\sqrt[3]{5^4}\right)^3\), which is equal to \(5^4\) by the definition of cube root. Cubing \(\left(\sqrt[3]{5}\right)^4\) also results in \(5^4\). To see this, note that \(\left(\left(\sqrt[3]{5}\right)^4\right)^3 =\left(\left(\sqrt[3]{5}\right)^3\right)^4=5^4\).)
4.3: Fractional Powers Are Just Numbers (20 minutes)
Activity
This activity connects the ideas of roots, rational exponents, graphs of exponential functions, and decimal approximations. Students graph \(y=2^x\) for a few integer values of \(x\), approximate a smooth curve passing through those points, and use that curve to estimate the value of \(2^x\) for various rational inputs. Students then test these estimates using their understanding of the connection between rational exponents and roots.
Launch
Arrange students in groups of 2–3. Encourage them to compare their answers with their group after each question.
Since the first problem in this activity was designed to be completed without technology, ask students to put away any devices until after they complete it.
Design Principle(s): Cultivate conversation
Supports accessibility for: Language; Socialemotional skills
Student Facing

Complete the table as much as you can without using a calculator. (You should be able to fill in three spaces.)
\(x\) 0 \(\frac13\) \(\frac23\) 1 \(\frac43\) \(\frac53\) 2 \(2^x\) (using exponents) \(2^0\) \(2^{\frac13}\) \(2^{\frac23}\) \(2^1\) \(2^{\frac43}\) \(2^{\frac53}\) \(2^2\) \(2^x\) (decimal approximation)  Plot the points that you filled in.
 Connect the points as smoothly as you can.

Use this graph of \(y=2^x\) to estimate the value of the other powers in the table, and write your estimates in the table.
 Plot the points that you filled in.

Let’s investigate \(2^{\frac13}\):
 Write \(2^{\frac13}\) using radical notation.
 What is \(\left( 2^{\frac13}\right)^3\)?
 Raise your estimate from the table of \(2^{\frac13}\) to the third power. What should it be? How close did you get?

Let’s investigate \(2^{\frac23}\):
 Write \(2^{\frac23}\) using radical notation.
 What is the value of \(\left( 2^{\frac23}\right)^3\)?
 Raise your estimate from the table of \(2^{\frac23}\) to the third power. What should it be? How close did you get?
Student Response
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Student Facing
Are you ready for more?
Answer these questions using the fact that \((1.26)^3 = 2.000376\).
 Explain why \(\sqrt[3]{2}\) is very close to \(1.26\). Is it larger or smaller than \(1.26\)?
 Is it possible to write \(\sqrt[3]{2}\) exactly with a finite decimal expansion? Explain how you know.
Student Response
For access, consult one of our IM Certified Partners.
Activity Synthesis
Display several student graphs of \(y=2^x\) for all to see and invite students to share how close their estimates for the cubes of \(2^{\frac13}\) and \(2^{\frac23}\) were. The important takeaway from this activity is that whether we write the number as \(2^{\frac23}\) or \(\sqrt[3]{2^2}\), the expression is just a number and its value can be approximated.
Lesson Synthesis
Lesson Synthesis
In this lesson, students extended the connection between roots and unit fraction exponents to describe other positive rational exponents in terms of roots. Building on this foundation, students can also generalize to fourth roots, fifth roots, and so on. Here are some questions for discussion:
 “What is the value of \(\sqrt[4]{9^2}\)?” (3, because \(\sqrt[4]{9^2}=9^{\frac24}=9^{\frac12}=\sqrt{9}=3\))
 “Express the number 16 as many ways as you can using exponents or roots. Think about how you might use fractions as exponents.” (Possible responses: \(4^2\), \(\left(\sqrt[3]{16}\right)^3\), \(2^4\), \(2^{\frac{12}{3}}\), \(\sqrt[3]{2^{12}}\), \(\sqrt[5]{2^{20}}\), and many more.)
4.4: Cooldown  Third It (5 minutes)
CoolDown
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Student Lesson Summary
Student Facing
Using exponent rules, we know \(3^{\frac14}\) is the same as \(\sqrt[4]{3}\) because \(\left(3^{\frac14}\right)^4 =3\). But what about \(3^{\frac54}\)?
Using exponent rules,
\(\displaystyle 3^{\frac54}=\left(3^5\right)^{\frac14}\)
which means that
\(\displaystyle 3^{\frac54}=\sqrt[4]{3^5}\)
Since \(3^5=243\), we could just write \(3^{\frac54}=\sqrt[4]{243} \).
Alternatively, we could express the fraction \(\frac54\) as \(\frac14 \boldcdot 5\) instead. Using exponent rules, we get
\(\displaystyle 3^{\frac54}= \left(3^{\frac14}\right)^5= \left(\sqrt[4]{3}\right)^5\)
Here are more examples of exponents that are fractions and their equivalents:
\(x\)  0  \(\frac13\)  \(\frac23\)  1  \(\frac43\)  \(\frac53\)  2 

\(5^x\) (using exponents)  \(5^0\)  \(5^{\frac13}\)  \(5^{\frac23}\)  \(5^1\)  \(5^{\frac43}\)  \(5^{\frac53}\)  \(5^2\) 
\(5^x\) (equivalent expression)  1  \(\sqrt[3]{5}\)  \(\sqrt[3]{5^2}\) or \(\sqrt[3]{25}\)  5  \(\sqrt[3]{5^4}\) or \(\sqrt[3]{625}\)  \(\sqrt[3]{5^5}\) or \(\sqrt[3]{3125}\)  25 