smc

If the sidelengths of the cubes are expressed as $a,b,$ and $c,$ then we can write three equations: $$ab=12,bc=8,ac=6.$$ The volume is $abc.$ Notice symmetry in the equations. We can find $abc$ my multiplying all the equations and taking the positive square root. $$\begin{array}{rl}(ab)(bc)(ac)& =(12)(8)(6)\\ a^2{b}^2{c}^2& =576\\ abc& =\boxed{24{\text{ in}}^3}\end{array}$$