Unit 7, Lesson 3
Powers of Powers of 10
Grade 8
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The purpose of this Warm-up is to introduce the idea of raising a value with an exponent to another power. Computing the volume of a cube whose side lengths are themselves powers of 10 introduces the basic structure of a power to a power, which will lead to a general exponent rule in later activities.
Monitor for students who use different strategies, such as counting zeros to keep track of place value or writing 10,000 as
Give students 1–2 minutes of quiet work time followed by a brief whole-class discussion.
What is the volume of a giant cube that measures 10,000 km on each side? Be prepared to explain your reasoning.
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The purpose of this discussion is to introduce the idea that
Invite previously identified students to share their strategies for computing the volume. If not brought up in students’ explanations, show students the strategy of re-writing 10,000 as a power of 10 and raising
MLR2: Collect and Display
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In this activity, students explore patterns to discover the property
Give students 1 minute of quiet think time to complete the first unfinished row in the table before inviting 1–2 students to share and explain their answers. When it is clear that students understand how to complete the table, explain to them that they can skip one entry in the table, but they have to be able to explain why they skipped it. Give students 5–6 minutes to complete the remaining questions before a whole-class discussion.
Complete the table to explore patterns in the exponents when raising a power of 10 to a power. You may skip a single box in the table, but if you do, be prepared to explain why you skipped it.
| expression | expanded | single power of 10 |
|---|---|---|
The goal of this discussion is to reinforce the exponent rule for raising powers of 10 to another power. Introduce and explain the visual display prepared earlier. This display should be kept visible to students throughout the remainder of the unit.
Continue to reinforce student understanding of this rule by writing out an expanded form of each expression when discussing the following questions:
In this lesson, we developed a rule for raising a power of 10 to another power: Taking a power of 10 and raising it to another power is the same as multiplying the exponents.
To understand this, take
This works for any power of 10 raised to another power. For example,
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