Given $x,y,z >0$ and $xy^2z^3 = 108 $, what is the minimum value of $x+y+z$? Given $x,y,z >0$ and $xy^2z^3 = 108 $, what is the minimum value of $x+y+z$ ?
This is a homework problem, so if someone could just give me an outline/hint of the method used to solve this it would be much appreciated. 
Thanks

By AM-GM Inequality,$$x+y+z = x+\frac{y}{2}+\frac{y}{2}+\frac{z}{3}+\frac{z}{3}+\frac{z}{3}\geq6\left(\frac{xy^2z^3}{2^2\cdot3^3}\right)^\frac{1}{6}=6\left(\frac{108}{2^2\cdot3^3}\right)^\frac{1}{6}=6$$
Equality holds when $x=\frac{y}{2}=\frac{z}{3}$, so plugging this into $xy^2z^3=108$ we know the equality occurs when $x=1, y=2,z=3$.
