geometry

volume of sphere  $=\,\frac43\,*\,\pi\,*\, (\text{radius of sphere})^3$

volume of hemisphere  $=\,\frac12\,*\,\frac43\,*\,\pi\,*\, (\text{radius of hemisphere})^3$

Plug in  $3\sqrt[3]2$  for the radius of the hemisphere.

volume of hemisphere  $=\,\frac12\,*\,\frac43\,*\,\pi \,*\,(3\sqrt[3]2)^3 \\~\\ =\,\frac23\,*\,\pi\,*\,3^3\,*\,\sqrt[3]2^3 \\~\\ =\,\frac23\,*\,\pi\,*\,27\,*\,2 \\~\\ =\,36\pi$

The hemisphere has the same volume as the sphere, so....

                         volume of sphere  =  volume of hemisphere

$\frac43\,*\,\pi\,*\, (\text{radius of sphere})^3\,=\,36\pi$

                                                                         Divide both sides by  pi .

$\frac43\,*\, (\text{radius of sphere})^3\,=\,36$

                                                                         Multiply both sides by  3/4  .

$(\text{radius of sphere})^3\,=\,27$

                                                                         Take the cube root of both sides.

$\text{radius of sphere}\,=\,3\,\text{ cm}$

Nice, hectictar.....!!!!