To solve the equation
\[ \frac{x}{18} = \frac{72}{x^3} \]
we can start by cross-multiplying to eliminate the fractions:
\[ x \cdot x^3 = 72 \cdot 18 \]
This simplifies to:
\[ x^4 = 1296 \]
Next, we take the fourth root of both sides:
\[ x = \pm \sqrt[4]{1296} \]
Now, we need to find \(\sqrt[4]{1296}\). First, find the prime factorization of 1296:
\[ 1296 = 36 \times 36 = 6^2 \times 6^2 = (6^2)^2 = 6^4 \]
So,
\[ 1296 = 6^4 \]
Therefore,
\[ \sqrt[4]{1296} = 6 \]
Thus, we get:
\[ x = \pm 6 \]
The final answer is:
\[ \boxed{\pm 6} \]