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?
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.
- How could you represent the large square as a power of 10?
- If each small square represents 102, then what does the medium rectangle represent? The large square?
- If the medium rectangle represents 105, then what does the large square represent? The small square?
- If the large square represents 10100, then what does the medium rectangle represent? The small square?
Lesson 2.3 Multiplying Powers of Ten
- 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?
- a. Use the patterns you found in the table to rewrite 10m · 10n as an equivalent expression with a single exponent, like 10x.
b. Use your rule to write 104 · 100 with a single exponent. What does this tell you about the value of 100?
- 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.)
- How many ways are there to make 106 by multiplying smaller powers of 10 together?
- How many ways are there to make 107 in the same way? 108?
Lesson 2 Practice Problems
- Write each expression with a single exponent:
- 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.
- Here is triangle ABC.
- 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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