Lesson 3

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

1. 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?
2. Use the patterns you found in the table to rewrite \(\left(10^m\right)^n\) as an equivalent expression with a single exponent, like \(10^{\boxed{\phantom{3}}}\).
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?

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.

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.