To determine which expression is equivalent to [tex]\(\frac{-9 x^{-1} y^{-9}}{-15 x^5 y^{-3}}\)[/tex], we will follow these steps:
1. Combine the coefficients:
The coefficients in the numerator and the denominator are [tex]\(-9\)[/tex] and [tex]\(-15\)[/tex] respectively.
[tex]\[
\frac{-9}{-15} = \frac{9}{15} = \frac{3}{5}
\][/tex]
2. Simplify the exponent of [tex]\(x\)[/tex]:
In the numerator, [tex]\(x\)[/tex] has an exponent of [tex]\(-1\)[/tex], and in the denominator, [tex]\(x\)[/tex] has an exponent of [tex]\(5\)[/tex]. When dividing, we subtract the exponent in the denominator from the exponent in the numerator:
[tex]\[
x^{-1 - 5} = x^{-6}
\][/tex]
3. Simplify the exponent of [tex]\(y\)[/tex]:
In the numerator, [tex]\(y\)[/tex] has an exponent of [tex]\(-9\)[/tex], and in the denominator, [tex]\(y\)[/tex] has an exponent of [tex]\(-3\)[/tex]. When dividing, we subtract the exponent in the denominator from the exponent in the numerator:
[tex]\[
y^{-9 - (-3)} = y^{-9 + 3} = y^{-6}
\][/tex]
Putting it all together, the expression simplifies to:
[tex]\[
\frac{3}{5 \cdot x^6 \cdot y^6}
\][/tex]
Thus, the equivalent expression is:
[tex]\[
\boxed{\frac{3}{5 x^6 y^6}}
\][/tex]
