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The given expression to simplify is:

\[
\frac{16}{2u^4 - 2v^4} \times \frac{u^8 - v^8}{14u^4 + 14v^4}
\]

### Step-by-Step Simplification:

1. **Factor the Denominators and Numerators**:
   - The denominator of the first fraction: 
     \[
     2u^4 - 2v^4 = 2(u^4 - v^4)
     \]
     And \(u^4 - v^4\) is a difference of squares:
     \[
     u^4 - v^4 = (u^2 - v^2)(u^2 + v^2)
     \]
     So:
     \[
     2u^4 - 2v^4 = 2(u^2 - v^2)(u^2 + v^2)
     \]
   
   - The denominator of the second fraction:
     \[
     14u^4 + 14v^4 = 14(u^4 + v^4)
     \]

2. **Simplify the Numerators**:
   - The numerator of the second fraction:
     \[
     u^8 - v^8 = (u^4 - v^4)(u^4 + v^4)
     \]
     Using the factorization of \(u^4 - v^4\) from the first step, we get:
     \[
     u^8 - v^8 = (u^2 - v^2)(u^2 + v^2)(u^4 + v^4)
     \]

3. **Simplify the Full Expression**:
   Now substitute all the factored forms into the expression:
   \[
   \frac{16}{2(u^2 - v^2)(u^2 + v^2)} \times \frac{(u^2 - v^2)(u^2 + v^2)(u^4 + v^4)}{14(u^4 + v^4)}
   \]
   
   - Cancel out \((u^2 - v^2)\), \((u^2 + v^2)\), and \((u^4 + v^4)\) from both the numerator and the denominator:
     \[
     \frac{16}{2} \times \frac{1}{14} = \frac{16}{28}
     \]
   
4. **Final Simplification**:
   Simplifying the fraction:
   \[
   \frac{16}{28} = \frac{4}{7}
   \]

### Final Answer:
\[
\boxed{\frac{4}{7}}
\]

Would you like more details, or have any questions?

Here are five related questions for further exploration:
1. How do you recognize the difference of squares when factoring?
2. Why is it important to cancel common terms in a rational expression?
3. Can the method used here apply to other polynomial expressions with higher powers?
4. How would the process change if we introduced more variables or higher-degree terms?
5. Can you think of a real-world scenario where simplifying such expressions is useful?

**Tip**: Always check if terms in both the numerator and denominator can be factored, as this helps with significant simplifications.

Perform the operation and simplify the following expression: \( \frac{16}{2u^4 - 2v^4} \times \frac{u^8 - v^8}{14u^4 + 14v^4} \)

Learn how to simplify the expression involving polynomial factorization and rational expressions. This problem applies the difference of squares to simplify both the numerator and denominator. Suitable for students in Grades 9-12.

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