# 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: Taking Powers of Powers of 10

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.

Row 1 expression expanded single power of 10
Row 2 $(10^3)^2$ $(10 \boldcdot 10 \boldcdot 10)(10 \boldcdot 10 \boldcdot 10)$ $10^6$
Row 3 $(10^2)^5$ $(10 \boldcdot 10)(10 \boldcdot 10)(10 \boldcdot 10)(10 \boldcdot 10)(10 \boldcdot 10)$
Row 4   $(10 \boldcdot 10 \boldcdot 10)(10 \boldcdot 10 \boldcdot 10)(10 \boldcdot 10 \boldcdot 10)(10 \boldcdot 10 \boldcdot 10)$
Row 5 $(10^4)^2$
Row 6 $(10^8)^{11}$
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.

## 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.