What is the volume, in cubic inches, of a rectangular box, whose faces have areas of $24$ square inches, $16$ square inches and $6$ square inches?
If $l$, $w$, and $h$ represent the dimensions of the rectangular box, we look for the volume $lwh$. We arbitrarily set $lw=24$, $wh=16$, and $lh=6$. Now notice that if we multiply all three equations, we get $l^2w^2h^2=24\cdot16\cdot6=2^3\cdot3\cdot2^4\cdot2\cdot3=2^8\cdot3^2$. To get the volume, we take the square root of each side and get $lwh=2^4\cdot3=\boxed{48}$ cubic inches.