Question:
A water-filled spherical balloon lands on a sidewalk, momentarily flattening to a hemisphere.  The ratio of the radius of the spherical balloon to the radius of the hemisphere can be expressed in the form $\sqrt[3]{a}$ for some real number $a$.  Compute $a$.

Answer:
Let the radius of the hemisphere be $r$.  The volume of a sphere with radius $r$ is $\frac{4}{3}\pi r^3$, so the volume of a hemisphere with radius $r$ is $\frac{2}{3}\pi r^3$.

Let the radius of the sphere be $x$, so we have \[\frac{4}{3}\pi x^3 = \frac{2}{3}\pi r^3.\]  Simplifying yields $2x^3=r^3$.  The desired ratio of the radius of the spherical balloon to the radius of the hemisphere is $\frac{x}{r}$.  We have \[\frac{x^3}{r^3} = \frac{1}{2}\] so \[\frac{x}{r} = \sqrt[3]{\frac{1}{2}}.\]  Thus, we see $a=\boxed{\frac{1}{2}}$.