The problem in the image asks you to simplify the expression for the cube of the volume of a right cone, given by $V = \frac{\pi r^2 h}{3}$. You are to find the simplified form of $V^3$, treating $\pi$ as a variable.

Step-by-Step Solution

1. Given Volume Formula: $V = \frac{\pi r^2 h}{3}$

2. Cube the Volume: To find $V^3$, you need to cube the entire expression: $V^3 = \left(\frac{\pi r^2 h}{3}\right)^3$

3. Apply the Cube: Apply the cube to both the numerator and the denominator: $V^3 = \frac{\left(\pi r^2 h\right)^3}{3^3}$

4. Simplify:

   • The numerator: $\left(\pi r^2 h\right)^3 = \pi^3 \times (r^2)^3 \times h^3 = \pi^3 r^6 h^3$
   • The denominator: $3^3 = 27$

   So the expression simplifies to: $V^3 = \frac{\pi^3 r^6 h^3}{27}$

Answer:

The correct simplified form of the expression is: $\frac{\pi^3 r^6 h^3}{27}$

Based on the options visible in the image, the correct choice would be the one matching this expression.

Would you like more details on this process, or do you have any other questions?

Here are some related questions you might consider:

1. What happens if you cube only the radius $r$ instead of the whole volume?
2. How would the expression change if you were asked to cube the square root of the volume?
3. Can you generalize this formula for a cone with a different shape?
4. What is the effect of changing the height $h$ on the volume's cube?
5. How does the unit of measurement affect the cubic volume calculation?

Tip: Always double-check the powers when working with exponents, as it's easy to make small mistakes that can lead to incorrect results!