Let's simplify $((4^4)/(16^4))^{-1/4}$ using the properties of rational exponents step by step:

Step 1: Simplify the base fraction $\frac{4^4}{16^4}$

We know that $16 = 4^2$. Thus, $16^4 = (4^2)^4 = 4^{8}$.

Substituting this: $\frac{4^4}{16^4} = \frac{4^4}{4^8}.$

Using the property of exponents $\frac{a^m}{a^n} = a^{m-n}$, we get: $\frac{4^4}{4^8} = 4^{4-8} = 4^{-4}.$

Step 2: Apply the exponent $-\frac{1}{4}$

The expression becomes: $(4^{-4})^{-\frac{1}{4}}.$

Using the property $(a^m)^n = a^{m \cdot n}$, we multiply the exponents: $4^{-4 \cdot (-\frac{1}{4})} = 4^{1}.$

Step 3: Simplify

$4^1 = 4.$

Final Answer:

$\boxed{4}$

---

Would you like a more detailed explanation or have any related questions? 😊

Here are 5 related questions to explore further:

1. What is the general rule for simplifying $(a^m)^n$?
2. How do negative exponents affect the value of a base?
3. What is the result of $((2^6)/(4^3))^{-1/2}$?
4. Can you derive a similar simplification if the base was not a power of another number?
5. How can you express roots (e.g., square roots or cube roots) using fractional exponents?

Tip: When dealing with rational exponents, always simplify the base first to make calculations easier.