To solve the equation:

$\frac{1}{x^2} - 4x^{\frac{1}{4}} = 0$

let's follow these steps:

1. Isolate terms: Move one term to the other side to make it easier to solve for $x$.

   $\frac{1}{x^2} = 4x^{\frac{1}{4}}$

2. Rewrite in terms of powers: Rewrite both terms in terms of powers of $x$.

   • The left side, $\frac{1}{x^2}$, is equivalent to $x^{-2}$.
   • The right side, $4x^{\frac{1}{4}}$, is already in terms of a power of $x$.

   So the equation becomes:

   $x^{-2} = 4x^{\frac{1}{4}}$

3. Rewrite equation by making the powers equal: Divide both sides by $x^{\frac{1}{4}}$ to express it as a single power of $x$.

   This yields:

   $x^{-\frac{9}{4}} = 4$

4. Solve for $x$: Rewrite $x^{-\frac{9}{4}} = 4$ in terms of radicals and solve for $x$ by taking appropriate powers.