Simplify the expression $\frac{x^4 \cdot (x^{-2}y^3)^{-3}}{2y \cdot 2xy^3}$.

1. Problem Description

Simplify the expression $\frac{x^4 \cdot (x^{-2}y^3)^{-3}}{2y \cdot 2xy^3}$.

2. Solution Steps

First, simplify the numerator:

$x^4 \cdot (x^{-2}y^3)^{-3} = x^4 \cdot (x^{-2})^{-3} \cdot (y^3)^{-3} = x^4 \cdot x^{(-2)(-3)} \cdot y^{3(-3)} = x^4 \cdot x^6 \cdot y^{-9} = x^{4+6} \cdot y^{-9} = x^{10} y^{-9}$.

Next, simplify the denominator:

$2y \cdot 2xy^3 = 2 \cdot 2 \cdot x \cdot y \cdot y^3 = 4xy^{1+3} = 4xy^4$.

Now, we can rewrite the entire expression as:

$\frac{x^{10} y^{-9}}{4xy^4}$.

Using the quotient rule for exponents: $\frac{x^a}{x^b} = x^{a-b}$ and $\frac{y^a}{y^b} = y^{a-b}$, we have:

$\frac{x^{10}}{x} \cdot \frac{y^{-9}}{y^4} \cdot \frac{1}{4} = \frac{x^{10-1}}{1} \cdot \frac{y^{-9-4}}{1} \cdot \frac{1}{4} = \frac{x^9}{1} \cdot \frac{y^{-13}}{1} \cdot \frac{1}{4} = \frac{x^9 y^{-13}}{4}$.

To eliminate the negative exponent, we write:

$\frac{x^9}{4y^{13}}$.

3. Final Answer

$\frac{x^9}{4y^{13}}$