A right, rectangular prism has three faces with areas of $6,8$ and $12$ square inches. What is the volume of the prism, in cubic inches?
To find the volume of a right, rectangular prism, I need to multiply the lengths of the three edges that meet at a vertex. I can use the fact that the area of a face is the product of the lengths of the two edges that form it. So, if I call the edge lengths $a,b,c$, then I have the equations $ab=6$, $ac=8$, and $bc=12$. I want to find $abc$, which is the volume of the prism. I notice that if I multiply the three equations together, I get $(ab)(ac)(bc) = (6)(8)(12)$. This simplifies to $(abc)^2 = 576$, so $abc = \sqrt{576}$. I can use a calculator or recall that $\sqrt{576} = 24$, since $24^2 = 576$. Therefore, the volume of the prism is 24 cubic inches. 