To simplify the given expression:

$\frac{a^2 - b^2}{a^4 - b^4}$

Step 1: Factorize both the numerator and the denominator.

• The numerator $a^2 - b^2$ is a difference of squares: $a^2 - b^2 = (a - b)(a + b)$

• The denominator $a^4 - b^4$ is also a difference of squares. It can be factored as: $a^4 - b^4 = (a^2 - b^2)(a^2 + b^2)$ Now substitute the factorized forms into the expression:

$\frac{(a - b)(a + b)}{(a^2 - b^2)(a^2 + b^2)}$

Step 2: Simplify by canceling $a^2 - b^2$.

Since $a^2 - b^2$ is common in both the numerator and denominator, we cancel it out:

$\frac{(a - b)(a + b)}{(a^2 + b^2)}$

Final Answer:

$\frac{(a - b)(a + b)}{a^2 + b^2}$

Let me know if you'd like a more detailed explanation of any step or if you have further questions!

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Related Questions:

1. How do you factorize higher-order polynomials like $a^6 - b^6$?
2. What are the special cases for factoring $a^2 - b^2$ when $a = b$?
3. Can you show this simplification using an example with specific numbers for $a$ and $b$?
4. What happens if $a^2 + b^2 = 0$? Can this expression be simplified further?
5. How does this problem relate to rational functions in algebra?

Tip: Always look for factoring opportunities in algebraic expressions to simplify them efficiently.