To solve the equation \( \frac{27}{x^2} = x \), we can start by multiplying both sides by \( x^2 \) to eliminate the fraction (assuming \( x \neq 0 \)):
\[ 27 = x^3 \]
Now, we solve for \( x \) by taking the cube root of both sides:
\[ x = \sqrt[3]{27} \]
Since \( 27 = 3^3 \), we have:
\[ x = 3 \]
Thus, the solution is:
\[ \boxed{3} \]