# Lesson 3

Exponents That Are Unit Fractions

## 3.1: Sometimes It’s Squared and Sometimes It’s Cubed (5 minutes)

### Warm-up

The purpose of this activity is to elicit strategies and understandings students have for square and cube roots. These understandings help students develop fluency and will be helpful later in this lesson when students will connect square and cube roots to exponents of \(\frac12\) and \(\frac13\). Students might note the negative solutions to the quadratic equations, but it is not necessary to emphasize those solutions at this time if they are not mentioned. Students will explore the symmetry of positive and negative solutions to quadratic equations more deeply in upcoming lessons.

While participating in this activity, students need to be precise in their word choice and use of language (MP6), because they have no algebraic method that they can use to solve and no calculators that could give approximate solutions numerically. They have to be able to recall the definitions of square and cube roots.

### Launch

Ask students to put away any devices. Tell students that their task is to find the most precise answers they can without a calculator.

### Student Facing

Find a solution to each equation.

- \(x^2 = 25\)
- \(z^2 = 7\)
- \(y^3 = 8\)
- \(w^3 = 19\)

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

Briefly discuss students’ responses. Here are some questions for discussion:

- How did you find the answers that are not integers? (I estimated a value for \(w\) and then cubed it to see how close it was to 19. I used a square root sign to write \(z\) as \(\sqrt7\).)
- How could we tell if our answers to these questions make sense? (We could square or cube them; for example, we know \(y\) is 2 because \(2^3=8\).)
- Could we write \(x\) or \(y\) using radicals? (Yes, because \(5=\sqrt{25}\) and \(2=\sqrt[3]8\).)

Key discussion points are that we can write exact answers using radical signs and that we can check the accuracy of our answers by squaring or cubing, as appropriate.

## 3.2: To the...Half? (15 minutes)

### Activity

The purpose of this activity is for students to connect numbers that are a base to the \(\frac12\) power to square roots. First, they use a graph to estimate the value of such numbers. Then, they extend exponent rules to find that the numbers must be square roots of the base. The first number students explore is \(9^{\frac12}\). They will find that this number squares to make 9, which immediately indicates that \(9^{\frac12}\) is either 3 or -3. It is likely that students will not bring up the possibility that \(9^{\frac12}=\text-3\) because the graph suggests \(9^{\frac12}\) should be positive. If students do not bring up the possibility that \(9^{\frac12}=\text-3\), it is not important to mention at this time. In an upcoming lesson, students will study pairs of positive and negative square roots more deeply. They will also be introduced to the convention that the square root of a number is the positive square root.

In the second part of the activity, students study \(3^{\frac12}\) and find that it squares to make 3. This is a less familiar square root, so students will not have an immediate number in mind. Rather, the wording of the task encourages students to think about what \(\left(3^{\frac12}\right)^2=3\) means in terms of roots.

Making graphing technology available gives students an opportunity to choose appropriate tools strategically (MP5).

### Launch

Tell students that they have worked with integer exponents, but now they will think about what it means to have an exponent of \(\frac12\).

*Action and Expression: Develop Expression and Communication.*Maintain a display of important terms and vocabulary. During the launch take time to review the exponent rules and the definition of square root (the square root of \(x \) is a number that squares to make \(x \)) and display them as a reference for students.

*Supports accessibility for: Memory; Language*

### Student Facing

- Clare said, “I know that \(9^2 = 9 \boldcdot 9\), \(9^1 = 9\), and \(9^0 = 1\). I wonder what \(9^{\frac12}\) means?” First, she graphed \(y = 9^x\) for some whole number values of \(x\), and estimated \(9^{\frac12}\) from the graph.
- Graph the function yourself. What estimate do you get for \(9^{\frac{1}{2}}\)?
- Using the properties of exponents, Clare evaluated \(9^{\frac12} \boldcdot 9^{\frac12}\). What did she get?
- For that to be true, what must the value of \(9^{\frac12}\) be?

- Diego saw Clare’s work and said, “Now I’m wondering about \(3^{\frac12}\).” First he graphed \(y = 3^x\) for some whole number values of \(x\), and estimated \(3^{\frac12}\) from the graph.
- Graph the function yourself. What estimate do you get for \(3^{\frac{1}{2}}\)?
- Next he used exponent rules to find the value of \(\left (3^{\frac12}\right)^2\). What did he find?
- Then he said, “That looks like a root!” What do you think he means?

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

Ask students to share their estimates for \(9^{\frac12}\) and \(3^{\frac12}\), and display their graphs for all to see. Ask students to share their responses to questions 2 and 3. The goal is for students to explain how applying exponent rules means that \(9^{\frac12}=3\). Then, ask students to share their responses to the last question. The important takeaway is that following the exponent rules means that \(b^\frac12\) is another way of writing \(\sqrt{b}\) because \(\left(b^{\frac12}\right)^2=b^{\frac12 \boldcdot 2} = b\).

*Speaking: MLR8 Discussion Supports.*As students share their response for solving the first question, press for details by asking how they know that \(9^{\frac{1}{2}}\) is equal to 3. Also ask students which properties of exponents they used to show that \(9^{\frac{1}{2}}\) is equal to 3. Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class. This will help students justify why \(b^{\frac{1}{2}}\) is another way of writing \(\sqrt{b}\).

*Design Principle(s): Support sense-making; Optimize output (for justification)*

## 3.3: Fraction of What, Exactly? (5 minutes)

### Activity

The purpose of this activity is for students to extend what they learned about square roots and \(\frac12\) exponents to cube roots and \(\frac13\) exponents.

### Launch

Tell students that they will use the same reasoning as in the previous activity to make sense of numbers to the \(\frac12\) power or \(\frac13\) power. Give a few minutes of quiet work time followed by a brief, whole-class discussion.

*Action and Expression: Internalize Executive Functions.*To support development of organizational skills, check in with students within the first 2–3 minutes of work time. Check that students are finding the exact values and remind them of how they could use the exponent rules to check their answers if needed. Invite students to show their thinking by rewriting the fractional exponents.

*Supports accessibility for: Memory; Organization*

### Student Facing

Use the exponent rules and your understanding of roots to find the exact value of:

- \(25^{\frac12}\)
- \(15^{\frac12}\)
- \(8^{\frac13}\)
- \(2^{\frac13}\)

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

The key takeaway is that extending exponent rules to these fractions means that \(b^{\frac12}=\sqrt{b}\) and \(b^{\frac13}=\sqrt[3]{b}\). Select students to share their responses for \(15^{\frac12}\) and \(2^{\frac13}\).

Conclude the discussion by reminding students that these expressions which involve the \(\sqrt{}\) symbol are often called *radical* expressions because the technical name for the symbol is the *radix* (which is also where the word root comes from). Expressions like \(\sqrt{15}\) and \(\sqrt[3]{2}\) are examples of radical expressions.

*Speaking: MLR8 Discussion Supports.*As students share their responses for \(15^{\frac{1}{2}}\) and \(2^{\frac{1}{3}}\), press for details by asking how they know that \(15^{\frac{1}{2}}\) is equal to \(\sqrt{15}\) and \(2^{\frac{1}{3}}\) is equal to \(\sqrt[3]{2}\). Also ask students which properties of exponents they used to show that the exponential and radical expressions are equivalent. Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class. This will help students justify why \(b^{\frac{1}{2}} = \sqrt{b}\) and \(b^{\frac{1}{3}} = \sqrt[3]{b}\).

*Design Principle(s): Support sense-making; Optimize output (for justification)*

## 3.4: Exponents and Radicals (10 minutes)

### Activity

In this activity, students match equivalent expressions involving exponents and radicals. This is the first time they see negative rational exponents, so expect a lot of discussion around those expressions. Students will have more chances to interpret negative exponents, so it is not crucial for them to show mastery at this time.

Monitor for groups who reason about the expressions with negative fractional exponents in different ways. For example, some students may see \(7^{\text- \frac12}\) as \(\left(7^{\text-1}\right)^{\frac12}\), while others will think of the expression as \(\left(7^{\frac12}\right)^{\text-1}\). Ask these groups to be ready to share their reasoning during the discussion.

### Launch

Arrange students in groups of 2. Give students a few minutes of quiet think time before asking them to share their responses with their partners. Tell students that if there is disagreement, they should work to reach agreement. Follow with a whole-class discussion.

*Conversing: MLR8 Discussion Supports.*Arrange students in groups of 2. Invite students to take turns finding a match and explaining their reasoning to their partner. Display the following sentence frames for all to see: “ ____ and ____ are equal because…” and “I noticed _____, so I matched….” Encourage students to challenge each other when they disagree. This will help students clarify their reasoning about equivalent expressions that involve exponents.

*Design Principle(s): Support sense-making; Maximize meta-awareness*

### Student Facing

Match each exponential expression to an equivalent expression.

- \(7^3\)
- \(7^2\)
- \(7^1\)
- \(7^0\)
- \(7^{\text-1}\)
- \(7^{\text-2}\)
- \(7^{\text-3}\)
- \(7^{\frac{1}{2}}\)
- \(7^{\text-\frac{1}{2}}\)
- \(7^{\frac13}\)
- \(7^{\text-\frac{1}{3}}\)

- \(\frac{1}{49}\)
- \(\frac{1}{343}\)
- \(\sqrt7\)
- \(\frac{1}{\sqrt[3]{7}}\)
- \(\sqrt[3]{7}\)
- 49
- \(\frac{1}{\sqrt7}\)
- 343
- 7
- \(\frac{1}{7}\)
- 1

### Student Response

For access, consult one of our IM Certified Partners.

### Student Facing

#### Are you ready for more?

How do we know without counting that the number of circles equals the number of squares? Because we can match every circle to exactly one square, and each square has a match:

We say that we have shown that there is a one-to-one correspondence of the set of circles and the set of squares. We can do this with infinite sets, too! For example, there are the same “number” of positive integers as there are even positive integers:

Every positive integer is matched to exactly one even positive integer, and every even integer has a match! We have shown that there is a one-to-one correspondence between the set of positive integers and the set of even positive integers. Whenever we can make a one-to-one matching like this of the positive integers to another set, we say the other set is *countable.*

- Show that the set of square roots of positive integers is countable.
- Show that the set of positive integer roots of 2 is countable.
- Show that the set of positive integer roots of positive integers is countable. (Hint: there is a famous proof that the positive rational numbers are countable. Find and study this proof.)

### Student Response

For access, consult one of our IM Certified Partners.

### Anticipated Misconceptions

If students aren’t sure how to match the negative rational exponents, ask them if they could make an educated guess based on the other patterns they notice.

### Activity Synthesis

Begin the discussion by selecting 2–3 groups to share how they matched the expressions with positive exponents. Next, invite previously identified groups to share how they matched the expressions with negative exponents. While students share their thinking, display any exponent rules referenced for all to see, such as \(\left(b^m\right)^n = b^{m \boldcdot n}\) or \(b^{\text-n} = \frac{1}{b^n}\). It is important for students to understand that fractional exponents follow all the same exponent rules as integer exponents.

## Lesson Synthesis

### Lesson Synthesis

In this lesson, students found that \(b^{\frac12}=\sqrt{b}\) and \(b^{\frac13}=\sqrt[3]{b}\). Here are some questions for discussion:

- “How would you explain the meaning of the \(\frac12\) exponent in \(3^{\frac12}\) to a student who is absent for this lesson so that they could understand the idea fully?” (Using exponent rules, squaring \(3^{\frac12}\) gives a value of 3 since \(\left(3^{\frac12}\right)^2 = 3^{\frac12 \boldcdot 2}=3\). Since \(3^{\frac12}\) is a positive number that squares to make 3, then \(3^{\frac12} = \sqrt{3}\).)
- “What would be another way to write \(13^{\frac15}\)? Explain your thinking.” (\(\sqrt[5]{13}\), because \(\left(13^{\frac15}\right)^5=13^{\frac15 \boldcdot 5}=13^1=13\).)

## 3.5: Cool-down - Write It Two Ways (5 minutes)

### Cool-Down

For access, consult one of our IM Certified Partners.

## Student Lesson Summary

### Student Facing

How can we make sense of the expression \(11^{\frac12}\)? For this expression to make any sense at all, we should be able to apply exponent rules to it. Let’s try squaring \(11^{\frac12}\) using exponent rules: \(\left ( 11^{\frac12}\right )^2 = 11^{\frac12 \boldcdot 2}\), which is simply 11. In other words, if we square the number \(11^{\frac12}\) using exponent rules, we get 11. That means that \(11^{\frac12}\) must be equal to \(\sqrt{11}\).

Similarly, \(11^{\frac13}\) must be equal to \(\sqrt[3]{11}\) because \(\begin{array}{} \left(11^{\frac13}\right)^3 &= 11^{\frac13 \boldcdot 3} \\ &=11 \end{array}\)

In general, if \(a\) is any positive number, then

\(\displaystyle a^{\frac12} = \sqrt{a}\)

and

\(\displaystyle a^{\frac13} = \sqrt[3]{a}\)

Remember, these expressions that involve the \(\sqrt{}\) symbol are often referred to as *radical* expressions.