To determine which expression is equivalent to [tex]\(\frac{-9 x^{-1} y^{-9}}{-15 x^5 y^{-3}}\)[/tex], follow these steps:
1. Simplify the numerical coefficient:
[tex]\[
\frac{-9}{-15} = \frac{9}{15} = \frac{3}{5}
\][/tex]
2. Simplify the [tex]\(x\)[/tex] terms:
- The term in the numerator [tex]\(x^{-1}\)[/tex] can be written as [tex]\(\frac{1}{x}\)[/tex].
- Therefore, the expression has [tex]\(\frac{1}{x}\)[/tex] in the numerator.
- In the denominator, there is [tex]\(x^5\)[/tex].
When simplifying [tex]\(x^{-1}\)[/tex] and [tex]\(x^5\)[/tex], the expression for [tex]\(x\)[/tex] becomes:
[tex]\[
x^{-1} \cdot x^{-5} = x^{-1 - 5} = x^{-6}
\][/tex]
Since [tex]\(x^{-6} = \frac{1}{x^6}\)[/tex], we need to recognize that it will be in the denominator as [tex]\(x^6\)[/tex].
3. Simplify the [tex]\(y\)[/tex] terms:
- The numerator has [tex]\(y^{-9}\)[/tex], which is [tex]\(\frac{1}{y^9}\)[/tex].
- The denominator has [tex]\(y^{-3}\)[/tex], which is [tex]\(\frac{1}{y^3}\)[/tex].
When simplifying [tex]\(y^{-9}\)[/tex] and [tex]\(y^{-3}\)[/tex], the expression for [tex]\(y\)[/tex] becomes:
[tex]\[
y^{-9} \cdot y^{3} = y^{-9 + 3} = y^{-6}
\][/tex]
Since [tex]\(y^{-6} = \frac{1}{y^6}\)[/tex], we know that [tex]\(y^6\)[/tex] will be in the denominator.
Combining everything, the simplified expression is:
[tex]\[
\frac{3}{5 x^6 y^6}
\][/tex]
Hence, 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]
