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.