To determine which expressions are equivalent to the given expression [tex]\( y^{-8} y^3 x^0 x^{-2} \)[/tex], let's simplify the expression step by step.
1. Simplify [tex]\( y \)[/tex] exponents:
[tex]\[
y^{-8} \cdot y^3 = y^{-8 + 3} = y^{-5}
\][/tex]
2. Simplify [tex]\( x \)[/tex] exponents:
[tex]\[
x^0 \cdot x^{-2} = x^{0 - 2} = x^{-2}
\][/tex]
Combining these results, we get:
[tex]\[
y^{-5} \cdot x^{-2} = x^{-2} \cdot y^{-5}
\][/tex]
Now, let's compare this simplified expression [tex]\( x^{-2} y^{-5} \)[/tex] with the given options:
1. [tex]\( y^{-24} \)[/tex]:
[tex]\[
y^{-24} \neq x^{-2} y^{-5}
\][/tex]
So, this is not equivalent.
2. [tex]\( x^2 y^{-11} \)[/tex]:
[tex]\[
x^2 y^{-11} \neq x^{-2} y^{-5}
\][/tex]
So, this is not equivalent.
3. [tex]\(\frac{x^2}{y^{11}}\)[/tex]:
[tex]\[
\frac{x^2}{y^{11}} = x^2 y^{-11} \neq x^{-2} y^{-5}
\][/tex]
So, this is not equivalent.
4. [tex]\( x^{-2} y^{-5} \)[/tex]:
[tex]\[
x^{-2} y^{-5} = x^{-2} y^{-5}
\][/tex]
This is equivalent.
5. [tex]\(\frac{1}{y^{24}} \)[/tex]:
[tex]\[
\frac{1}{y^{24}} = y^{-24} \neq x^{-2} y^{-5}
\][/tex]
So, this is not equivalent.
6. [tex]\(\frac{1}{x^2 4 y^5} \)[/tex]:
[tex]\[
\frac{1}{x^2 4 y^5} = \frac{1}{4} \cdot \frac{1}{x^2 y^5} \neq x^{-2} y^{-5}
\][/tex]
So, this is not equivalent.
Hence, the correct equivalent expressions are:
[tex]\[
x^{-2} y^{-5}
\][/tex]
So, the correct answers are:
[tex]\[
x^{-2} y^{-5}
\][/tex]
