Lesson 3Powers of Powers of 10

Learning Goal

Let’s look at powers of powers of 10.

Learning Targets

I can explain and use a rule for raising a power of 10 to a power.

Lesson Terms

base (of an exponent)

Warm Up: Big Cube

What is the volume of a giant cube that measures 10,000 km on each side?

Activity 1: Taking Powers of Powers of 10

Problem 1

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
${(10^{3})}^{2}$	$(10 \cdot 10 \cdot 10) (10 \cdot 10 \cdot 10)$	$10^{6}$
${(10^{2})}^{5}$	$(10 \cdot 10) (10 \cdot 10) (10 \cdot 10) (10 \cdot 10) (10 \cdot 10)$
	$(10 \cdot 10 \cdot 10) (10 \cdot 10 \cdot 10) (10 \cdot 10 \cdot 10) (10 \cdot 10 \cdot 10)$
${(10^{4})}^{2}$
${(10^{8})}^{11}$

If you chose to skip one entry in the table, which entry did you skip? Why?

Problem 2

Use the patterns you found in the table to rewrite ${(10^{m})}^{n}$ as an equivalent expression with a single exponent, like $10^{}$ .

Problem 3

If you took the amount of oil consumed in 2 months in 2013 worldwide, you could make a cube of oil that measures $10^{3}$ meters on each side. How many cubic meters of oil is this? Do you think this would be enough to fill a pond, a lake, or an ocean?

Activity 2: How Do the Rules Work?

Andre and Elena want to write $10^{2} \cdot 10^{2} \cdot 10^{2}$ with a single exponent.

Andre says, “When you multiply powers with the same base, it just means you add the exponents, so $10^{2} \cdot 10^{2} \cdot 10^{2} = 10^{2 + 2 + 2} = 10^{6}$ .”
Elena says, “ $10^{2}$ is multiplied by itself 3 times, so $10^{2} \cdot 10^{2} \cdot 10^{2} = {(10^{2})}^{3} = 10^{2 + 3} = 10^{5}$ .”

Do you agree with either of them? Explain your reasoning.

Are you ready for more?

$2^{12} = 4, 096$ . How many other whole numbers can you raise to a power and get $4, 096$ ? Explain or show your reasoning.

Lesson Summary

In this lesson, we developed a rule for taking 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.

See what happens when raising $10^{4}$ to the power of 3. ${(10^{4})}^{3} = 10^{4} \cdot 10^{4} \cdot 10^{4} = 10^{12}$

This works for any power of powers of 10. For example, ${(10^{6})}^{11} = 10^{66}$ . This is another rule that will make it easier to work with and make sense of expressions with exponents.