To simplify the expression [tex]\(\frac{-9 x^{-1} y^{-9}}{-15 x^5 y^{-3}}\)[/tex]:
1. Simplify the coefficients:
The coefficients are [tex]\(-9\)[/tex] and [tex]\(-15\)[/tex]:
[tex]\[
\frac{-9}{-15} = \frac{9}{15} = \frac{3}{5}
\][/tex]
2. Simplify the [tex]\(x\)[/tex] terms:
The [tex]\(x\)[/tex] terms are [tex]\(x^{-1}\)[/tex] in the numerator and [tex]\(x^5\)[/tex] in the denominator:
[tex]\[
\frac{x^{-1}}{x^5} = x^{-1 - 5} = x^{-6}
\][/tex]
3. Simplify the [tex]\(y\)[/tex] terms:
The [tex]\(y\)[/tex] terms are [tex]\(y^{-9}\)[/tex] in the numerator and [tex]\(y^{-3}\)[/tex] in the denominator:
[tex]\[
\frac{y^{-9}}{y^{-3}} = y^{-9 - (-3)} = y^{-9 + 3} = y^{-6}
\][/tex]
4. Combine the results:
Putting all the simplified parts together, we get:
[tex]\[
\frac{3}{5} x^{-6} y^{-6}
\][/tex]
To write it in the standard form, we recognize that [tex]\(x^{-6}\)[/tex] is [tex]\(\frac{1}{x^6}\)[/tex] and [tex]\(y^{-6}\)[/tex] is [tex]\(\frac{1}{y^6}\)[/tex]:
[tex]\[
\frac{3}{5} \cdot \frac{1}{x^6} \cdot \frac{1}{y^6} = \frac{3}{5 x^6 y^6}
\][/tex]
Thus, the expression [tex]\(\frac{-9 x^{-1} y^{-9}}{-15 x^5 y^{-3}}\)[/tex] simplifies to [tex]\(\frac{3}{5 x^6 y^6}\)[/tex].
The correct answer is:
[tex]\[
\boxed{\frac{3}{5 x^6 y^6}}
\][/tex]
