Let's break down the given expression step-by-step and identify which of the provided expressions are equivalent.
Given:
[tex]\[ y^{-8} y^3 x^0 x^{-2} \][/tex]
1. Simplify each component:
- [tex]\( y^{-8} \)[/tex] remains [tex]\( y^{-8} \)[/tex].
- [tex]\( y^3 \)[/tex] stays [tex]\( y^3 \)[/tex].
- [tex]\( x^0 \)[/tex] is equal to 1.
- [tex]\( x^{-2} \)[/tex] stays [tex]\( x^{-2} \)[/tex].
2. Combine the [tex]\( y \)[/tex] terms:
[tex]\[ y^{-8} \cdot y^3 = y^{-8 + 3} = y^{-5} \][/tex]
3. Now, combine the result with the [tex]\( x \)[/tex] term:
[tex]\[ y^{-5} \cdot x^{-2} = x^{-2} y^{-5} \][/tex]
So, the simplified expression is:
[tex]\[ x^{-2} y^{-5} \][/tex]
Next, let's check the provided expressions to see which are equivalent:
1. [tex]\( x^2 y^{-11} \)[/tex]:
[tex]\[ x^2 y^{-11} \neq x^{-2} y^{-5} \][/tex]
2. [tex]\( \frac{x^2}{y^{12}} \)[/tex]:
[tex]\[ \frac{x^2}{y^{12}} = x^2 y^{-12} \neq x^{-2} y^{-5} \][/tex]
3. [tex]\( y^{-24} \)[/tex]:
[tex]\[ y^{-24} \neq x^{-2} y^{-5} \][/tex]
4. [tex]\( \frac{1}{y^{24}} \)[/tex]:
[tex]\[ \frac{1}{y^{24}} = y^{-24} \neq x^{-2} y^{-5} \][/tex]
5. [tex]\( \frac{1}{x^2 y^5} \)[/tex]:
[tex]\[ \frac{1}{x^2 y^5} = x^{-2} y^{-5} \][/tex]
6. [tex]\( x^{-2} y^{-5} \)[/tex]:
[tex]\[ x^{-2} y^{-5} = x^{-2} y^{-5} \][/tex]
Thus, the expressions that are equivalent to the given expression [tex]\( y^{-8} y^3 x^0 x^{-2} \)[/tex] are:
[tex]\[ \frac{1}{x^2 y^5} \text{ and } x^{-2} y^{-5} \][/tex]
So, the correct answers are:
[tex]\[ \frac{1}{x^2 y^5} \][/tex] and [tex]\[ x^{-2} y^{-5} \][/tex]
