To find the scale factor of the smaller cylinder to the larger cylinder based on their volumes, we can first calculate the ratio of their volumes.
The volume of the larger cylinder is \( 512\pi \) in\(^3\), and the volume of the smaller cylinder is \( 64\pi \) in\(^3\). We can set up the ratio of the volumes as follows:
\[ \frac{V_{\text{small}}}{V_{\text{large}}} = \frac{64\pi}{512\pi} \]
The \(\pi\) cancels out:
\[ \frac{64}{512} \]
Now, simplify this fraction:
\[ \frac{64 \div 64}{512 \div 64} = \frac{1}{8} \]
The volumes of similar shapes are related to the cube of the scale factor. If \(k\) is the scale factor from the smaller cylinder to the larger cylinder, then:
\[ k^3 = \frac{V_{\text{small}}}{V_{\text{large}}} = \frac{1}{8} \]
To find \(k\), we take the cube root of both sides:
\[ k = \sqrt[3]{\frac{1}{8}} = \frac{1}{2} \]
Therefore, the scale factor of the small cylinder to the larger cylinder is:
\[ \frac{1}{2} \]
So, the final answer is that the scale factor of the small cylinder to the larger cylinder is \(\frac{1}{2}\).