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.

Practice Problems ▶