The volume \( V \) of a right cone is given by the formula:
\[ V = \frac{1}{3} \pi r^2 h \]
To find the cube of the volume \( V^3 \), we will elevate the entire volume expression to the third power:
\[ V^3 = \left(\frac{1}{3} \pi r^2 h\right)^3 \]
When we expand this expression, we will cube each component inside the parentheses:
\[ V^3 = \left(\frac{1}{3}\right)^3 \cdot (\pi)^3 \cdot (r^2)^3 \cdot (h)^3 \]
Calculating each piece:
- \(\left(\frac{1}{3}\right)^3 = \frac{1}{27}\)
- \((\pi)^3 = \pi^3\)
- \((r^2)^3 = r^{2 \cdot 3} = r^6\)
- \((h)^3 = h^3\)
Putting this all together, we get:
\[ V^3 = \frac{1}{27} \pi^3 r^6 h^3 \]
So, the simplified form of the expression that represents the cube of the volume \( V^3 \) is:
\[ V^3 = \frac{1}{27} \pi^3 r^6 h^3 \]
8 answers