Lesson 5

Negative Rational Exponents

5.1: Math Talk: Don’t Be Negative (5 minutes)

Warm-up

This Math Talk encourages students to think about exponent rules and to rely on properties of exponents to mentally solve problems. The understandings elicited here will be helpful later in the lesson when students use graphs to approximate the value of $$2^x$$ for various negative rational exponents.

The final question is the first time students see an exponent that is a negative fraction. To use their prior knowledge to evaluate this expression, students need to look for and make use of structure (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 whole-class discussion.

Representation: Internalize Comprehension. To support working memory, provide students with sticky notes or mini whiteboards.
Supports accessibility for: Memory; Organization

Student Facing

Evaluate mentally.

$$9^2$$

$$9^{\text-2}$$

$$9^{\frac12}$$

$$9^{\text-\frac12}$$

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 $$\underline{\hspace{.5in}}$$’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 $$\underline{\hspace{.5in}}$$’s strategy?”
• “Do you agree or disagree? Why?”

If this strategy is not mentioned, consider asking students how they would use (91/2)-1 or (9-1)1/2 to evaluate the last expression. If the results of the pre-assessment showed that students need extra practice with exponent rules, highlight any student strategies that leverage these rules and carefully record student thinking for all to see.

Speaking: MLR8 Discussion Supports. Display sentence frames to support students when they explain their strategy. For example, "First, I _____ because…” or "I noticed _____ so I….” Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class.
Design Principle(s): Optimize output (for explanation)

5.2: Negative Fractional Powers Are Just Numbers (15 minutes)

Activity

In a previous lesson, students graphed $$y=9^x$$ for integer values of $$x$$ and then interpolated the value of $$9^{\frac12}$$. In this activity, students use the same technique for approximating the value of $$2^x$$ for various negative rational exponents. Students test those 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 completing each of the 3 questions.

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.

Engagement: Develop Effort and Persistence. Encourage and support opportunities for peer collaboration. When students share their work with a partner, display sentence frames to support conversation such as: “First, I _____ because. . .”, “How did you get . . .?”, “That could/couldn’t be true because . . . .”
Supports accessibility for: Language; Social-emotional skills

Student Facing

1. Complete the table as much as you can without using a calculator. (You should be able to fill in three spaces.)

 $$x$$ $$2^x$$ (using exponents) $$2^x$$ (decimal approximation) -2 $$\text{-}\frac53$$ $$\text{-}\frac43$$ -1 $$\text{-}\frac23$$ $$\text{-}\frac13$$ 0 $$2^{\text- 2}$$ $$2^{\text{-}\frac53}$$ $$2^{\text{-}\frac43}$$ $$2^{\text- 1}$$ $$2^{\text{-}\frac23}$$ $$2^{\text{-}\frac13}$$ $$2^0$$
1. Plot these powers of 2 in the coordinate plane. ​​​​​​
2. Connect the points as smoothly as you can.
3. Use your graph of $$y=2^x$$ to estimate the value of the other powers in the table, and write your estimates in the table.
2. Let’s investigate $$2^{\text{-} \frac13}$$.

1. Write $$2^{\text{-} \frac13}$$ using radical notation.
2. What is the value of $$\left( 2^{\text{-} \frac13}\right)^3$$?
3. Raise your estimate of $$2^{\text{-} \frac13}$$ to the third power. What should it be? How close did you get?
3. Let’s investigate $$2^{\text{-} \frac23}$$.

1. Write $$2^{\text{-} \frac23}$$ using radical notation.
2. What is $$\left( 2^{\text{-} \frac23}\right)^3$$?
3. Raise your estimate of $$2^{\text{-} \frac23}$$ to the third power. What should it be? How close did you get?

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 $$2^{\text-\frac13}$$ and $$2^{\text-\frac23}$$ were. The important takeaway for this activity is that whether we write the number as $$2^{\text- \frac23}$$ or $$\sqrt[3]{2^{\text-2}}$$ or $$\frac{1}{\sqrt[3]{4}}$$, the expression is just a number and its value can be approximated.

Representing, Conversing: MLR7 Compare and Connect. Use this routine to prepare students for the whole-class discussion about the graph of $$y=2^x$$. After students complete the table and sketch the graph of $$y=2^x$$, invite them to create a visual display of their work. Then ask students to quietly circulate and observe at least two other visual displays in the room. Give students quiet think time to consider what is the same and what is different about their tables and graphs. Next, ask students to find a partner to discuss what they noticed. Listen for and amplify the language students use to compare and contrast the estimates for $$2^x$$ for various values of $$x$$ between -2 and 0.
Design Principle(s): Cultivate conversation

5.3: Any Fraction Can Be an Exponent (15 minutes)

Activity

In this activity, students practice identifying expressions involving rational exponents with equivalent radical expressions.

Launch

Since this activity was designed to be completed without technology, ask students to put away any devices until after they complete it.

Student Facing

1. For each set of 3 numbers, cross out the expression that is not equal to the other two expressions.
1. $$8^{\frac45}$$, $$\sqrt[4]{8}^5$$, $$\sqrt[5]{8}^4$$
2. $$8^{\text{-} \frac45}$$, $$\dfrac{1}{\sqrt[5]{8^4}}$$, $$\text-\dfrac{1}{\sqrt[5]{8^4}}$$
3. $$\sqrt{4^3}$$, $$4^{\frac32}$$, $$4^{\frac23}$$
4. $$\dfrac{1}{\sqrt{4^3}}$$, $$\text-4^{\frac32}$$, $$4^{\text-\frac32}$$
2. For each expression, write an equivalent expression using radicals.
1. $$17^{\frac32}$$
2. $$31^{\text{-} \frac32}$$
3. For each expression, write an equivalent expression using only exponents.
1. $$\left(\sqrt{3}\right)^4$$
2. $$\dfrac{1}{\left(\sqrt[3]{5}\right)^6}$$

Student Facing

Write two different expressions that involve only roots and powers of 2 which are equivalent to $$\frac{4^\frac23}{8^\frac14}$$.

Anticipated Misconceptions

If students have trouble getting started, remind them of previous strategies they have used or seen to deal with rational exponents. They could try breaking fractions apart into multiples of unit fractions. Or they could strategically raise numbers to powers in order to see if they are the same. For example, $$4^{\frac23}$$ cubed is $$4^2$$, so if it’s the same as $$4^{\frac32}$$, then cubing that should also give us $$4^2$$, but it doesn’t. Therefore, they’re not the same.

If students aren’t sure how to read the notation $$\sqrt[4]{\phantom{33}}$$ and $$\sqrt[5]{\phantom{33}}$$, tell them that these are fourth and fifth roots, and they work the same way as square and cube roots: if you raise $$\sqrt[4]{10}$$ to the fourth power, you get 10, and if you raise $$\sqrt[5]{10}$$ to the fifth power, you get 10.

Activity Synthesis

Once students have finished, ask them to use a calculator to check if each answer they chose or wrote is equal to the given expression. You may need to demonstrate how they can enter roots into the calculator.

5.4: Make These Exponents Less Complicated (10 minutes)

Optional activity

This activity is optional because it goes beyond the depth of understanding required to address the standards.

Students build on their work in the previous activity to compare expressions with rational exponents.

Launch

Arrange students in groups of 2. Give students quiet work time before asking them to compare how they matched the expressions with their partner. Follow with a brief, whole-class discussion.

Engagement: Develop Effort and Persistence. Encourage and support opportunities for peer interactions. Prior to the whole-class discussion, invite students to share their work with a partner. Display sentence frames to support student conversation such as: “One thing that is the same is. . .”, “I noticed _____ so I . . .”, “Why did you . . .?”, “I agree/disagree because . . . .”
Supports accessibility for: Language; Social-emotional skills

Student Facing

Match expressions into groups according to whether they are equal. Be prepared to explain your reasoning.

$$\left(\sqrt{3}\right)^4$$

$$\sqrt{3^2}$$

$$\left (3^{\frac12}\right )^4$$

$$(\sqrt{3})^2 \boldcdot (\sqrt{3})^2$$

$$\left (3^2 \right) ^{\frac12}$$

$$3^2$$

$$3^{\frac42}$$

$$\left (3^{\frac12}\right)^2$$

Activity Synthesis

Select a group to share how they matched the expressions and explain their reasoning. Ask other students whether they agree or disagree. As a class, work to reach agreement.

Speaking, Representing: MLR8 Discussion Supports. Use this routine to support whole-class discussion. After each group shares, provide the class with the following sentence frames to help them respond: "I agree because . . .” or "I disagree because . . . .” If necessary, revoice student ideas to demonstrate mathematical language use by restating a statement as a question in order to clarify, apply appropriate language, and involve more students. For example, a statement such as, “This is equal to 9,” can be restated as a question, “Is the square root of 3 raised to the fourth power equal to 9?”
Design Principle(s): Support sense-making

Lesson Synthesis

Lesson Synthesis

In this lesson, students have worked with expressions that involve negative rational exponents. Here are some questions for discussion:

• “How are $$2^{\text- \frac12}$$ and $$2^{\frac12}$$ alike? How are they different?” (They are alike because they both involve an exponent of $$\frac12$$, which means the expression can be described using square roots. They are different because the negative exponent means that they are reciprocals of each other. In particular, $$2^{\frac12}=\sqrt{2}$$ while $$2^{\text- \frac12}=\frac{1}{2^{\frac12}}=\frac{1}{\sqrt{2}}$$.)
• “How could you write $$5^{\text- \frac{2}{3}}$$ using radicals?” $$\left(\dfrac{1}{\sqrt[3]{25}}\right)$$
• “What would be the result of cubing $$5^{\text- \frac{2}{3}}$$?” (Cubing would give $$5^{\text-2}$$, which is $$\frac{1}{25}$$.)

Student Lesson Summary

Student Facing

When we have a number with a negative exponent, it just means we need to find the reciprocal of the number with the exponent that has the same magnitude, but is positive. Here are two examples:

$$\displaystyle 7^{\text{-} 5} = \dfrac{1}{7^5}$$

$$\displaystyle 7^{\text{-} \frac65} = \dfrac{1}{7^{\frac65}}$$

The table shows a few more examples of exponents that are fractions and their radical equivalents.

 $$x$$ $$5^x$$ (using exponents) $$5^x$$ (equivalent expressions) -1 $$\text{-} \frac23$$ $$\text{-} \frac13$$ 0 $$\frac13$$ $$\frac23$$ 1 $$5^{\text-1}$$ $$5^{\text{-} \frac23}$$ $$5^{\text{-} \frac13}$$ $$5^0$$ $$5^{\frac13}$$ $$5^{\frac23}$$ $$5^1$$ $$\frac15$$ $$\dfrac{1}{\sqrt[3]{5^2}}$$ or $$\dfrac{1}{\sqrt[3]{25}}$$ $$\dfrac{1}{\sqrt[3]{5}}$$ 1 $$\sqrt[3]{5}$$ $$\sqrt[3]{5^2}$$ or $$\sqrt[3]{25}$$ 5