# Lesson 3

Powers of Powers of 10

Let's look at powers of powers of 10.

### 3.1: Big Cube

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

### 3.2: Raising Powers of 10 to Another Power

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
$$(10^3)^2$$ $$(10 \boldcdot 10 \boldcdot 10)(10 \boldcdot 10 \boldcdot 10)$$ $$10^6$$
$$(10^2)^5$$ $$(10 \boldcdot 10)(10 \boldcdot 10)(10 \boldcdot 10)(10 \boldcdot 10)(10 \boldcdot 10)$$
$$(10 \boldcdot 10 \boldcdot 10)(10 \boldcdot 10 \boldcdot 10)(10 \boldcdot 10 \boldcdot 10)(10 \boldcdot 10 \boldcdot 10)$$
$$(10^4)^2$$
$$(10^8)^{11}$$
2. If you chose to skip one entry in the table, which entry did you skip? Why?
1. Use the patterns you found in the table to rewrite $$\left(10^m\right)^n$$ as an equivalent expression with a single exponent, like $$10^{\boxed{\phantom{3}}}$$.
2. 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?

### 3.3: How Do the Rules Work?

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

• Andre says, “When you multiply powers with the same base, it just means you add the exponents, so $$10^2 \boldcdot 10^2 \boldcdot 10^2 = 10^{2+2+2} = 10^6$$.”

• Elena says, “$$10^2$$ is multiplied by itself 3 times, so $$10^2 \boldcdot 10^2 \boldcdot 10^2 = (10^2)^3 = 10^{2+3} = 10^5$$.”

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

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

### 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.

$$\left(10^4\right)^3 =10^4 \boldcdot 10^4 \boldcdot 10^4 = 10^{12}$$

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

### Glossary Entries

• base (of an exponent)

In expressions like $$5^3$$ and $$8^2$$, the 5 and the 8 are called bases. They tell you what factor to multiply repeatedly. For example, $$5^3$$ = $$5 \boldcdot 5 \boldcdot 5$$, and $$8^2 = 8 \boldcdot 8$$.