A triangle has angles of $30^\circ$ and $45^\circ$. If the side opposite the $45^\circ$ angle has length $8$, then the side opposite the $30^\circ$ angle has length

$\textbf{(A) }4\qquad\textbf{(B) }4\sqrt{2}\qquad\textbf{(C) }4\sqrt{3}\qquad\textbf{(D) }4\sqrt{6}\qquad \textbf{(E) }6$
1. **Determine the third angle of the triangle:**
   Given two angles of the triangle are \(30^\circ\) and \(45^\circ\). The sum of the angles in any triangle is \(180^\circ\). Therefore, the third angle is:
   \[
   180^\circ - 30^\circ - 45^\circ = 105^\circ
   \]

2. **Drop an altitude from the \(105^\circ\) angle:**
   Dropping an altitude from the \(105^\circ\) angle to the side opposite it divides the triangle into two right triangles: a \(45^\circ-45^\circ-90^\circ\) triangle and a \(30^\circ-60^\circ-90^\circ\) triangle.

3. **Analyze the \(30^\circ-60^\circ-90^\circ\) triangle:**
   In a \(30^\circ-60^\circ-90^\circ\) triangle, the sides are in the ratio \(1 : \sqrt{3} : 2\). The side opposite the \(30^\circ\) angle is half the hypotenuse. Given that the side opposite the \(45^\circ\) angle (which is the hypotenuse of the \(30^\circ-60^\circ-90^\circ\) triangle) is \(8\), the side opposite the \(30^\circ\) angle is:
   \[
   \frac{8}{2} = 4
   \]

4. **Analyze the \(45^\circ-45^\circ-90^\circ\) triangle:**
   In a \(45^\circ-45^\circ-90^\circ\) triangle, the sides are in the ratio \(1 : 1 : \sqrt{2}\). The side opposite the \(30^\circ\) angle in the original triangle is the hypotenuse of this \(45^\circ-45^\circ-90^\circ\) triangle. Therefore, the length of the side opposite the \(30^\circ\) angle is:
   \[
   4 \times \sqrt{2} = 4\sqrt{2}
   \]

Conclusion:
\[
\boxed{4\sqrt{2}}
\]