Given the expression [tex]\(\frac{2}{x^0 - y^3}\)[/tex], let's begin by simplifying it:
1. Simplify the base expression:
[tex]\[
x^0 = 1 \quad \text{for any } x
\][/tex]
Therefore,
[tex]\[
\frac{2}{x^0 - y^3} = \frac{2}{1 - y^3}
\][/tex]
Now, we will check each given expression to see if they are equivalent to [tex]\(\frac{2}{1 - y^3}\)[/tex].
2. Check each given expression:
A. [tex]\(\frac{1}{\left(x^2 - y^2\right)} \cdot \frac{1}{\left(x^2 + y^2\right)}\)[/tex]
This can be simplified to:
[tex]\[
\frac{1}{(x^2 - y^2)(x^2 + y^2)}
\][/tex]
Simplifying further,
[tex]\[
(x^2 - y^2)(x^2 + y^2) = x^4 - y^4
\][/tex]
Therefore,
[tex]\[
\frac{1}{(x^2 - y^2)(x^2 + y^2)} = \frac{1}{x^4 - y^4}
\][/tex]
This is not equivalent to [tex]\(\frac{2}{1 - y^3}\)[/tex].
B. [tex]\(\frac{2}{(x^2)^2 - (y^2)^1}\)[/tex]
Simplify the expression in the denominator:
[tex]\[
(x^2)^2 = x^4 \quad \text{and} \quad (y^2)^1 = y^2
\][/tex]
Hence,
[tex]\[
\frac{2}{x^4 - y^2}
\][/tex]
This is not equivalent to [tex]\(\frac{2}{1 - y^3}\)[/tex].
C. [tex]\(\frac{2}{x^2 - y^3} \cdot \frac{1}{x^3 - y^2}\)[/tex]
This can be written as:
[tex]\[
\frac{2}{(x^2 - y^3)(x^3 - y^2)}
\][/tex]
This does not simplify in any way that is equivalent to [tex]\(\frac{2}{1 - y^3}\)[/tex].
D. [tex]\(\frac{2}{x^3 - y^3} \cdot \frac{1}{x^3 + y^3}\)[/tex]
This can be written as:
[tex]\[
\frac{2}{(x^3 - y^3)(x^3 + y^3)}
\][/tex]
Simplifying the denominator,
[tex]\[
(x^3 - y^3)(x^3 + y^3) = x^6 - y^6
\][/tex]
Hence,
[tex]\[
\frac{2}{x^6 - y^6}
\][/tex]
This is also not equivalent to [tex]\(\frac{2}{1 - y^3}\)[/tex].
3. Conclusion:
None of the provided expressions [tex]\(A, B, C,\)[/tex] or [tex]\(D\)[/tex] are equivalent to [tex]\(\frac{2}{1 - y^3}\)[/tex]. Therefore, no expressions apply.
