https://file.aiquickdraw.com/mathbot/file/1725383938407nnlbslse.jpg

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.

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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!

The volume of a right cone is defined as V = πr²h/3. What is the simplified form of the expression that represents the cube of this volume, V³? Treat the constant π as you would a variable in your answer.

Learn how to simplify the expression for the cube of the volume of a right cone using algebraic techniques. This problem involves key concepts in exponents and volume calculation, suitable for Grades 8-10.

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