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.