To simplify the given expression [tex]\(\frac{-9 x^{-1} y^{-9}}{-15 x^5 y^{-3}}\)[/tex], follow these steps:
1. Simplify the numerical coefficients:
Consider the coefficients [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:
Combine the exponents of [tex]\(x\)[/tex] in the numerator and the denominator:
[tex]\[
\frac{x^{-1}}{x^5} = x^{-1 - 5} = x^{-6}
\][/tex]
3. Simplify the [tex]\(y\)[/tex] terms:
Combine the exponents of [tex]\(y\)[/tex] in the numerator and the denominator:
[tex]\[
\frac{y^{-9}}{y^{-3}} = y^{-9 - (-3)} = y^{-9 + 3} = y^{-6}
\][/tex]
4. Combine everything into the simplified expression:
Combine the simplified coefficient with the simplified [tex]\(x\)[/tex] and [tex]\(y\)[/tex] terms:
[tex]\[
\frac{3}{5} \cdot x^{-6} \cdot y^{-6} = \frac{3}{5 x^6 y^6}
\][/tex]
So, the expression equivalent to [tex]\(\frac{-9 x^{-1} y^{-9}}{-15 x^5 y^{-3}}\)[/tex] is:
[tex]\[
\boxed{\frac{3}{5 x^6 y^6}}
\][/tex]
