To determine which expression is equivalent to [tex]\(\frac{-9 x^{-1} y^{-9}}{-15 x^5 y^{-3}}\)[/tex], we need to simplify this fraction step-by-step.
### Step 1: Simplify the Coefficient
Firstly, let's simplify the numerical part of the fraction [tex]\(\frac{-9}{-15}\)[/tex]:
[tex]\[
\frac{-9}{-15} = \frac{9}{15} = \frac{3}{5}
\][/tex]
So the coefficient simplifies to [tex]\(\frac{3}{5}\)[/tex].
### Step 2: Simplify the [tex]\(x\)[/tex] Terms
Next, we simplify the [tex]\(x\)[/tex] terms in the numerator and the denominator. We have [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]
### Step 3: Simplify the [tex]\(y\)[/tex] Terms
Then, we simplify the [tex]\(y\)[/tex] terms in the numerator and the denominator. We have [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]
### Step 4: Combine Results
Having simplified the coefficients and the variables, we combine them:
[tex]\[
\frac{-9 x^{-1} y^{-9}}{-15 x^5 y^{-3}} = \frac{3}{5} x^{-6} y^{-6}
\][/tex]
Or in a more standard form, using positive exponents:
[tex]\[
\frac{3}{5 x^6 y^6}
\][/tex]
### Conclusion
Thus, the given 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 choice is:
[tex]\[
\boxed{\frac{3}{5 x^6 y^6}}
\][/tex]
