Let's simplify the given expression step by step.

The expression is:

$\left(\frac{4x^4}{y^3}\right)^{-4}$

Step 1: Apply the exponent rule

When an entire fraction is raised to a negative exponent, we can invert the fraction and raise it to the positive exponent:

$\left(\frac{4x^4}{y^3}\right)^{-4} = \left(\frac{y^3}{4x^4}\right)^4$

Step 2: Apply the exponent to both the numerator and the denominator

We raise both the numerator and the denominator to the power of 4:

$\frac{(y^3)^4}{(4x^4)^4} = \frac{y^{12}}{4^4 \cdot x^{16}}$

Step 3: Simplify the constants and exponents

Now simplify the constant $4^4$ and leave the exponents as they are:

$4^4 = 256$

Thus, the simplified expression is:

$\frac{y^{12}}{256x^{16}}$

Final Answer

The simplified result is:

$\boxed{\frac{y^{12}}{256x^{16}}}$

This matches the second option in your list of answers.

Would you like a more detailed explanation on any part? Here are five related questions to deepen your understanding:

1. What happens if the exponent was positive instead of negative in the original expression?
2. How do you handle negative exponents when they apply to only one term?
3. How does raising a fraction to a power affect both the numerator and the denominator?
4. What is the significance of simplifying constants like $4^4$?
5. How does the distributive property of exponents work in such scenarios?

Tip: Remember, when simplifying expressions with exponents, carefully apply the rules for powers of powers and negative exponents!