47. Let three positive real numbers $x, y, z$. Try to find the maximum value of the algebraic expression $\frac{16 x+9 \sqrt{2 x y}+9 \sqrt[3]{3 x y z}}{x+2 y+z}$. (2006 Henan Province Mathematics Competition Problem)
47. Since
$$\begin{array}{l}
\frac{16 x+9 \sqrt{2 x y}+9 \sqrt[3]{3 x y z}}{x+2 y+z}= \\
\frac{16 x+\frac{9 \sqrt{x \cdot 18 y}}{3}+\frac{3 \sqrt[3]{x \cdot 18 y \cdot 36 z}}{3}}{x+2 y+z} \leqslant \\
\frac{16 x+\frac{3(x+18 y)}{2}+\frac{x+18 y+36 z}{2}}{x+2 y+z}=18
\end{array}$$

Therefore, the equality holds if and only if $x: y: z=36: 2: 1$.