Answer :
We are given the expression [tex]\(\left(\frac{x^{-4} y}{x^{-9} y^5}\right)^{-2}\)[/tex] and need to find an equivalent expression.
First, let's simplify the expression inside the parentheses:
1. Simplify the [tex]\(x\)[/tex]-terms:
[tex]\[ \frac{x^{-4}}{x^{-9}} = x^{-4 - (-9)} = x^{-4 + 9} = x^5 \][/tex]
2. Simplify the [tex]\(y\)[/tex]-terms:
[tex]\[ \frac{y}{y^5} = y^{1 - 5} = y^{-4} \][/tex]
After simplifying the terms, we are left with:
[tex]\[ \left(x^5 y^{-4}\right)^{-2} \][/tex]
Next, we apply the negative exponent -2 to both the [tex]\(x\)[/tex]-term and the [tex]\(y\)[/tex]-term:
1. Apply [tex]\(-2\)[/tex] to [tex]\(\bm{x^5}\)[/tex]:
[tex]\[ (x^5)^{-2} = x^{5 \times (-2)} = x^{-10} \][/tex]
2. Apply [tex]\(-2\)[/tex] to [tex]\(\bm{y^{-4}}\)[/tex]:
[tex]\[ (y^{-4})^{-2} = y^{-4 \times (-2)} = y^8 \][/tex]
Putting these parts together, we get:
[tex]\[ x^{-10} y^8 \][/tex]
This can be rewritten as:
[tex]\[ \frac{y^8}{x^{10}} \][/tex]
Therefore, the expression equivalent to [tex]\(\left(\frac{x^{-4} y}{x^{-9} y^5}\right)^{-2}\)[/tex] is:
[tex]\[ \boxed{\frac{y^8}{x^{10}}} \][/tex]
Thus, the correct answer is the first option:
[tex]\(\frac{y^8}{x^{10}}\)[/tex].
First, let's simplify the expression inside the parentheses:
1. Simplify the [tex]\(x\)[/tex]-terms:
[tex]\[ \frac{x^{-4}}{x^{-9}} = x^{-4 - (-9)} = x^{-4 + 9} = x^5 \][/tex]
2. Simplify the [tex]\(y\)[/tex]-terms:
[tex]\[ \frac{y}{y^5} = y^{1 - 5} = y^{-4} \][/tex]
After simplifying the terms, we are left with:
[tex]\[ \left(x^5 y^{-4}\right)^{-2} \][/tex]
Next, we apply the negative exponent -2 to both the [tex]\(x\)[/tex]-term and the [tex]\(y\)[/tex]-term:
1. Apply [tex]\(-2\)[/tex] to [tex]\(\bm{x^5}\)[/tex]:
[tex]\[ (x^5)^{-2} = x^{5 \times (-2)} = x^{-10} \][/tex]
2. Apply [tex]\(-2\)[/tex] to [tex]\(\bm{y^{-4}}\)[/tex]:
[tex]\[ (y^{-4})^{-2} = y^{-4 \times (-2)} = y^8 \][/tex]
Putting these parts together, we get:
[tex]\[ x^{-10} y^8 \][/tex]
This can be rewritten as:
[tex]\[ \frac{y^8}{x^{10}} \][/tex]
Therefore, the expression equivalent to [tex]\(\left(\frac{x^{-4} y}{x^{-9} y^5}\right)^{-2}\)[/tex] is:
[tex]\[ \boxed{\frac{y^8}{x^{10}}} \][/tex]
Thus, the correct answer is the first option:
[tex]\(\frac{y^8}{x^{10}}\)[/tex].
