jmc

If $l$, $w$, and $h$ represent the dimensions of the rectangular prism, we look for the volume $lwh$. We arbitrarily set $lw=6$, $wh=8$, and $lh=12$. Now notice that if we multiply all three equations, we get $l^2{w}^2{h}^2=6\cdot 8\cdot 12=3\cdot 2\cdot 2^3\cdot 2^2\cdot 3=2^6\cdot 3^2$. To get the volume, we take the square root of each side and get $lwh=2^3\cdot 3=\boxed{24}$ cubic inches.