a cube has a side length of 3x^6 units. a smaller cube has a side length of x^2 units. how many smaller…

Explanation:

Step1: Find volume of larger cube

The volume formula for a cube is $V = s^3$, where $s$ is the side - length. For the larger cube with side - length $s_1=3x^{6}$, its volume $V_1=(3x^{6})^3$. Using the power - of - a - product rule $(ab)^n=a^n b^n$, we have $V_1 = 3^3\times(x^{6})^3=27x^{18}$.

Step2: Find volume of smaller cube

For the smaller cube with side - length $s_2 = x^{2}$, its volume $V_2=(x^{2})^3$. Using the power - of - a - power rule $(a^m)^n=a^{mn}$, we get $V_2=x^{6}$.

Step3: Calculate number of smaller cubes

To find the number of smaller cubes that fit into the larger cube, we divide the volume of the larger cube by the volume of the smaller cube. Let $n$ be the number of smaller cubes. Then $n=\frac{V_1}{V_2}=\frac{27x^{18}}{x^{6}}$. Using the quotient rule $\frac{a^m}{a^n}=a^{m - n}$, we have $n = 27x^{18-6}=27x^{12}$.

Answer:

$27x^{12}$