Let's analyze each of the given expressions to determine if they are equivalent to the expression [tex]\( y^{-8} y^3 x^0 x^{-2} \)[/tex].
First, simplify the given expression [tex]\( y^{-8} y^3 x^0 x^{-2} \)[/tex]:
1. Combine the powers of [tex]\( y \)[/tex]:
[tex]\[
y^{-8} \cdot y^3 = y^{-8 + 3} = y^{-5}
\][/tex]
2. Combine the powers of [tex]\( x \)[/tex]:
[tex]\[
x^0 \cdot x^{-2} = x^{0-2} = x^{-2}
\][/tex]
So, the given expression simplifies to:
[tex]\[
y^{-5} \cdot x^{-2} = x^{-2} y^{-5}
\][/tex]
Next, let's check the given expressions one by one:
1. [tex]\( y^{-24} \)[/tex]:
[tex]\[
\text{This is not equivalent because it does not account for } x \text{ or combine to } y^{-5}.
\][/tex]
It is incorrect.
2. [tex]\( \frac{x^2}{y^{11}} \)[/tex]:
[tex]\[
\text{This expression cannot be reformulated to } x^{-2} y^{-5}.
\][/tex]
It is incorrect.
3. [tex]\( \frac{1}{x^2 y^5} \)[/tex]:
[tex]\[
\frac{1}{x^2 y^5} \text{ is equivalent to } x^{-2} y^{-5} \text{, which matches our simplified expression}.
\][/tex]
It is correct.
4. [tex]\( x^2 y^{-11} \)[/tex]:
[tex]\[
\text{This has different exponents of } x \text{ and } y \text{.}
\][/tex]
It is incorrect.
5. [tex]\( \frac{1}{y^{24}} \)[/tex]:
[tex]\[
\text{This does not have the } x^{-2} \text{ term and does not combine to } y^{-5}.
\][/tex]
It is incorrect.
6. [tex]\( x^{-2} y^{-5} \)[/tex]:
[tex]\[
\text{This is exactly our simplified expression.}
\][/tex]
It is correct.
Therefore, the correct answers are:
[tex]\[
\boxed{\frac{1}{x^2 y^5}, \quad x^{-2} y^{-5}}
\][/tex]
