((4^(4))/(16^(4)))^(-1/4) use the properties of rational exponents 

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.

((4^(4))/(16^(4)))^(-1/4) use the properties of rational exponents

Learn how to simplify the expression ((4^4)/(16^4))^(-1/4) using properties of rational exponents step by step. This solution demonstrates exponent rules like division and negative exponents. Suitable for students in Grades 8-10.

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