Let's simplify the given expression step-by-step:
The initial expression is:
[tex]\[ y^{-8} y^3 x^0 x^{-2} \][/tex]
1. Simplify the [tex]\(y\)[/tex] terms:
Combine the exponents of [tex]\(y\)[/tex]:
[tex]\[ y^{-8} \times y^3 = y^{-8+3} = y^{-5} \][/tex]
2. Simplify the [tex]\(x\)[/tex] terms:
Combine the exponents of [tex]\(x\)[/tex]:
[tex]\[ x^0 \times x^{-2} = x^{0+(-2)} = x^{-2} \][/tex]
So, the simplified expression is:
[tex]\[ x^{-2} y^{-5} \][/tex]
Next, let's compare this simplified expression with the given options to see which ones are equivalent:
1. [tex]\(x^2 y^{-11}\)[/tex]
[tex]\[ x^2 y^{-11} \][/tex] is not the same as [tex]\(x^{-2} y^{-5}\)[/tex].
2. [tex]\(\frac{x^2}{y^{11}}\)[/tex]
[tex]\[ \frac{x^2}{y^{11}} = x^2 y^{-11} \][/tex] is not the same as [tex]\(x^{-2} y^{-5}\)[/tex].
3. [tex]\(x^{-2} y^{-5}\)[/tex]
[tex]\[ x^{-2} y^{-5} \][/tex] is exactly the same as [tex]\(x^{-2} y^{-5}\)[/tex].
4. [tex]\(\frac{1}{x^2 y^5}\)[/tex]
Note that:
[tex]\[ \frac{1}{x^2 y^5} = x^{-2} y^{-5} \][/tex]
This matches our simplified expression [tex]\(x^{-2} y^{-5}\)[/tex].
5. [tex]\(y^{-24}\)[/tex]
[tex]\[ y^{-24} \][/tex] is not the same as [tex]\(x^{-2} y^{-5}\)[/tex].
6. [tex]\(\frac{1}{y^{2x}}\)[/tex]
[tex]\[ \frac{1}{y^{2x}} \][/tex] does not resemble [tex]\(x^{-2} y^{-5}\)[/tex].
Therefore, the correct equivalent expressions are:
- [tex]\(x^{-2} y^{-5}\)[/tex]
- [tex]\(\frac{1}{x^2 y^5}\)[/tex]
So, the correct answers are:
[tex]\[ \text{Option 3: } x^{-2} y^{-5} \][/tex]
[tex]\[ \text{Option 4: } \frac{1}{x^2 y^5} \][/tex]
