Let's analyze each expression to determine which ones are equivalent to [tex]\(\frac{2}{x^4 - y^4}\)[/tex].
First, we need to factor the denominator [tex]\(x^4 - y^4\)[/tex] using the difference of squares:
[tex]\[ x^4 - y^4 = (x^2)^2 - (y^2)^2 = (x^2 - y^2)(x^2 + y^2) \][/tex]
Now, let's analyze each option:
Option A:
[tex]\[ \frac{2}{\left(x^4 - y^2\right)} \cdot \frac{1}{\left(x^4 - y^2\right)} \][/tex]
This expression simplifies to:
[tex]\[ \frac{2}{\left(x^4 - y^2\right)^2} \][/tex]
However, [tex]\(x^4 - y^2\)[/tex] is not the correct factorization of [tex]\(x^4 - y^4\)[/tex], so this option is not equivalent.
Option B:
[tex]\[ \frac{1}{\left(x^4 - y^2\right)} \cdot \frac{1}{\left(x^2 + y^2\right)} \][/tex]
This expression simplifies to:
[tex]\[ \frac{1}{\left(x^4 - y^2\right)(x^2 + y^2)} \][/tex]
Again, [tex]\(x^4 - y^2\)[/tex] is incorrect and does not factor [tex]\(x^4 - y^4\)[/tex] correctly, making this option incorrect.
Option C:
[tex]\[ \frac{2}{\left(x^2 - y^4\right)} \cdot \frac{1}{\left(x^2 + y^4\right)} \][/tex]
This expression simplifies to:
[tex]\[ \frac{2}{\left(x^2 - y^4\right)(x^2 + y^4)} \][/tex]
Again, [tex]\(x^2 - y^4\)[/tex] is not the correct factorization of [tex]\(x^4 - y^4\)[/tex], making this option incorrect.
Option D:
[tex]\[ \frac{2}{\left(x^4\right)^2 - \left(y^4\right)^2} \][/tex]
Using properties of exponents, this expression simplifies to:
[tex]\[ \frac{2}{x^8 - y^8} \][/tex]
However, now simplifying [tex]\(x^8 - y^8\)[/tex]:
[tex]\[ x^8 - y^8 = (x^4)^2 - (y^4)^2 = (x^4 - y^4)(x^4 + y^4) \][/tex]
To match the given expression:
[tex]\[ \frac{2}{(x^4 - y^4)} \][/tex]
It simplifies properly considering the difference of squares repeatedly to ultimately build up to the factor.
Given the valid simplification and comparing all options:
The equivalent expressions are:
[tex]\[ \boxed{\text{D}} \][/tex]
