3.1: Big Cube
What is the volume of a giant cube that measures 10,000 km on each side?
Let's look at powers of powers of 10.
What is the volume of a giant cube that measures 10,000 km on each side?
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} |
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.
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.