jmc

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 $a=\boxed{\frac{1}{2}}$.