To solve the equation
\[ \frac{4^{1/4}}{x} = 4^{3/4}, \]
we can start by isolating \(x\). First, rearrange the equation by multiplying both sides by \(x\):
\[ 4^{1/4} = 4^{3/4} \cdot x. \]
Next, divide both sides by \(4^{3/4}\):
\[ x = \frac{4^{1/4}}{4^{3/4}}. \]
Using the laws of exponents, specifically \(a^m / a^n = a^{m-n}\), we can simplify the right hand side:
\[ x = 4^{1/4 - 3/4} = 4^{-2/4} = 4^{-1/2}. \]
Next, we can express \(4^{-1/2}\) in a more recognizable format. Since \(4 = 2^2\), we can rewrite it as:
\[ 4^{-1/2} = (2^2)^{-1/2} = 2^{-2/2} = 2^{-1} = \frac{1}{2}. \]
Thus, the solution for \(x\) is:
\[ x = \frac{1}{2}. \]