To determine the correct equivalent expressions for [tex]\( y^{-8} y^3 x^0 x^{-2} \)[/tex], let's simplify the given expression step by step.
1. Simplify the [tex]\( y \)[/tex] terms:
[tex]\[
y^{-8} \times y^3 = y^{(-8 + 3)} = y^{-5}
\][/tex]
2. Simplify the [tex]\( x \)[/tex] terms:
[tex]\[
x^0 \times x^{-2} = x^{(0 + (-2))} = x^{-2}
\][/tex]
3. Combine the simplified [tex]\( y \)[/tex] and [tex]\( x \)[/tex] terms:
[tex]\[
y^{-5} \times x^{-2}
\][/tex]
So, the simplified expression is:
[tex]\[
x^{-2} y^{-5}
\][/tex]
Next, we compare this with the given answer choices:
- [tex]\( y^{-24} \)[/tex]: This is not equivalent to [tex]\( x^{-2} y^{-5} \)[/tex].
- [tex]\(\frac{x^2}{y^{12}}\)[/tex]: This is not equivalent to [tex]\( x^{-2} y^{-5} \)[/tex].
- [tex]\(\frac{1}{y^{2 x}}\)[/tex]: This is not equivalent to [tex]\( x^{-2} y^{-5} \)[/tex].
- [tex]\(\frac{1}{x^2 y^5}\)[/tex]: This is equivalent to [tex]\( x^{-2} y^{-5} \)[/tex], because:
[tex]\[
\frac{1}{x^2 y^5} = x^{-2} y^{-5}
\][/tex]
- [tex]\(x^2 y^{-11}\)[/tex]: This is not equivalent to [tex]\( x^{-2} y^{-5} \)[/tex].
- [tex]\(x^{-2} y^{-5}\)[/tex]: This is directly equivalent to [tex]\( x^{-2} y^{-5} \)[/tex].
Therefore, the correct equivalent expressions are:
[tex]\[
\frac{1}{x^2 y^5} \quad \text{and} \quad x^{-2} y^{-5}
\][/tex]
