To simplify the given expression [tex]\(\frac{r^{-4} s^5}{r^2 s}\)[/tex], we can use the laws of exponents.
Starting with the expression:
[tex]\[
\frac{r^{-4} s^5}{r^2 s}
\][/tex]
1. Separate the expression into parts involving [tex]\(r\)[/tex] and [tex]\(s\)[/tex] respectively:
[tex]\[
\frac{r^{-4}}{r^2} \cdot \frac{s^5}{s}
\][/tex]
2. Simplify the expression involving [tex]\(r\)[/tex]:
When dividing exponents with the same base, subtract the exponents:
[tex]\[
\frac{r^{-4}}{r^2} = r^{-4 - 2} = r^{-6}
\][/tex]
3. Simplify the expression involving [tex]\(s\)[/tex]:
Similarly, for the [tex]\(s\)[/tex] terms:
[tex]\[
\frac{s^5}{s} = s^{5 - 1} = s^4
\][/tex]
4. Combine the simplified parts:
[tex]\[
r^{-6} \cdot s^4
\][/tex]
5. Express [tex]\(r^{-6}\)[/tex] as a positive exponent:
[tex]\[
r^{-6} = \frac{1}{r^6}
\][/tex]
6. Combine [tex]\(s^4\)[/tex] with [tex]\(\frac{1}{r^6}\)[/tex]:
[tex]\[
r^{-6} \cdot s^4 = \frac{s^4}{r^6}
\][/tex]
Therefore, the simplified expression is:
[tex]\[
\frac{s^4}{r^6}
\][/tex]
The correct answer is option C:
[tex]\[
\boxed{\frac{s^4}{r^6}}
\][/tex]
