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


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 \((10^3)^2\) \((10 \boldcdot 10 \boldcdot 10)(10 \boldcdot 10 \boldcdot 10)\) \(10^6\) \((10^2)^5\) \((10 \boldcdot 10)(10 \boldcdot 10)(10 \boldcdot 10)(10 \boldcdot 10)(10 \boldcdot 10)\) \((10 \boldcdot 10 \boldcdot 10)(10 \boldcdot 10 \boldcdot 10)(10 \boldcdot 10 \boldcdot 10)(10 \boldcdot 10 \boldcdot 10)\) \((10^4)^2\) \((10^8)^{11}\)  If you chose to skip one entry in the table, which entry did you skip? Why?

 Use the patterns you found in the table to rewrite \(\left(10^m\right)^n\) as an equivalent expression with a single exponent, like \(10^{\boxed{\phantom{3}}}\).
 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 \(8^2 = 8 \boldcdot 8\).