Simplify the expression $\frac{2x^3y^{-4}}{2yx^4 \cdot 2x^{-4}y^{-1}}$.

1. Problem Description

Simplify the expression $\frac{2x^3y^{-4}}{2yx^4 \cdot 2x^{-4}y^{-1}}$.

2. Solution Steps

First, simplify the denominator by multiplying the terms:

$2yx^4 \cdot 2x^{-4}y^{-1} = 2 \cdot 2 \cdot x^4 \cdot x^{-4} \cdot y \cdot y^{-1} = 4x^{4+(-4)}y^{1+(-1)} = 4x^0y^0 = 4(1)(1) = 4$.

So the expression becomes:

$\frac{2x^3y^{-4}}{4}$.

Now, we can simplify the fraction by dividing the coefficient:

$\frac{2x^3y^{-4}}{4} = \frac{1}{2}x^3y^{-4}$.

We can write the expression without negative exponents:

$\frac{1}{2}x^3y^{-4} = \frac{x^3}{2y^4}$.

3. Final Answer

$\frac{x^3}{2y^4}$