What is the simplified form of the expression (2x^(-2) * y^3)^(-2)/(2xzy^4 * x^(2)y^(-2) * y^4)?

To simplify the expression ((2x^{-2} y^3)^{-2}/(2xzy^4 \cdot x^{2}y^{-2} \cdot y^{4})), we can break it down step by step.

First, simplify the numerator, which is ((2x^{-2} y^3)^{-2}). When we raise a product to a power, we apply the exponent to each factor:

[

(2x^{-2}y^3)^{-2} = 2^{-2} (x^{-2})^{-2} (y^{3})^{-2} = \frac{1}{4} x^{4} y^{-6}

]

So, the numerator becomes (\frac{1}{4} x^{4} y^{-6}).

Next, let's simplify the denominator, which is (2xyz^{4} \cdot x^{2}y^{-2} \cdot y^{4}). We can combine the terms:

[

2xzy^4 \cdot x^2y^{-2} = 2x^{1 + 2}y^{4 - 2}z = 2x^{