To solve the equation \( \frac{27}{x^2} = x \), we can start by multiplying both sides by \( x^2 \) to eliminate the fraction:
\[ 27 = x^3 \]
Next, we can solve for \( x \) by taking the cube root of both sides:
\[ x = \sqrt[3]{27} \]
Since \( 27 = 3^3 \), we find:
\[ x = 3 \]
Thus, the solution is:
\[ \boxed{3} \]