Illustrative Mathematics Grade 8, Unit 7, Lesson 2: Multiplying Powers of Ten

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Illustrative Math
Grade 8

Lesson 2: Multiplying Powers of Ten

Let’s explore patterns with exponents when we multiply powers of 10.

Illustrative Math Unit 8.7, Lesson 2 (printable worksheets)

Lesson 2 Summary

In this lesson, we developed a rule for multiplying powers of 10: multiplying powers of 10 corresponds to adding the exponents together. To see this, multiply 10⁵ and 10². We know that 10⁵ has five factors that are 10 and 10² has two factors that are 10. That means that 10⁵ · 10² has 7 factors that are 10.

10⁵ · 10² = (10 · 10 · 10 · 10 · 10) · (10 · 10) = 10⁷

This will work for other powers of 10 too. So 10¹⁴ · 10⁴⁷ = 10⁶¹.

This rule makes it easier to understand and work with expressions that have exponents.

Lesson 2.1 100, 1, or 1/100?

Clare said she sees 100.
Tyler says he sees 1.
Mai says she sees 1/100.
Who do you agree with?

• See Video for Whole Lesson

Lesson 2.2 Picture a Power of 10

In the diagram, the medium rectangle is made up of 10 small squares. The large square is made up of 10 medium rectangles.

1. How could you represent the large square as a power of 10?
2. If each small square represents 10², then what does the medium rectangle represent? The large square?
3. If the medium rectangle represents 10⁵, then what does the large square represent? The small square?
4. If the large square represents 10¹⁰⁰, then what does the medium rectangle represent? The small square?

Lesson 2.3 Multiplying Powers of Ten

1. a. Complete the table to explore patterns in the exponents when multiplying powers of 10. You may skip a single box in the table, but if you do, be prepared to explain why you skipped it.
   b. If you chose to skip one entry in the table, which entry did you skip? Why?
2. a. Use the patterns you found in the table to rewrite 10^m · 10ⁿ as an equivalent expression with a single exponent, like 10^x.
   b. Use your rule to write 10⁴ · 10⁰ with a single exponent. What does this tell you about the value of 10⁰?
3. The state of Georgia has roughly 10⁷ human residents. Each human has roughly 10¹³ bacteria cells in his or her digestive tract. How many bacteria cells are there in the digestive tracts of all the humans in Georgia?

Are you ready for more?

There are four ways to make by multiplying smaller, positive powers of 10.
10¹ · 10¹ · 10¹ · 10¹
10¹ · 10¹ · 10²
10¹ · 10³
10² · 10²
(This list is complete if you don’t pay attention to the order you write them in. For example, we are only counting 10¹ · 10³ and 10³ · 10¹ once.)

1. How many ways are there to make 10⁶ by multiplying smaller powers of 10 together?
   • Show Answer

     10 ways

2. How many ways are there to make 10⁷ in the same way? 10⁸?
   • Show Answer

     14 ways
     21 ways

Lesson 2 Practice Problems

1. Write each expression with a single exponent:
2. A large rectangular swimming pool is 1,000 feet long, 100 feet wide, and 10 feet deep. The pool is filled to the top with water.
   a. What is the area of the surface of the water in the pool?
   b. How much water does the pool hold?
   c. Express your answers to the previous two questions as powers of 10.
3. Here is triangle ABC.
4. Elena and Jada distribute flyers for different advertising companies. Elena gets paid 65 cents for every 10 flyers she distributes, and Jada gets paid 75 cents for every 12 flyers she distributes.
   Draw graphs on the coordinate plane representing the total amount each of them earned, y, after distributing x flyers. Use the graph to decide who got paid more after distributing 14 flyers.

The Open Up Resources math curriculum is free to download from the Open Up Resources website and is also available from Illustrative Mathematics.

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