We are asked to simplify the expression $(4x)^{-4}$.

1. Problem Description

We are asked to simplify the expression $(4x)^{-4}$.

2. Solution Steps

We can use the property that $(ab)^n = a^n b^n$. Therefore,

$(4x)^{-4} = 4^{-4} x^{-4}$

We also know that $a^{-n} = \frac{1}{a^n}$. Using this, we have:

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

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

Substituting these back into our expression, we get:

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

Now, we calculate $4^4$.

$4^4 = 4 \cdot 4 \cdot 4 \cdot 4 = 16 \cdot 16 = 256$

So, we have:

$\frac{1}{4^4 x^4} = \frac{1}{256x^4}$

3. Final Answer

The simplified expression is $\frac{1}{256x^4}$.