To simplify the expression [tex]\(\left(\frac{-5 r^2 s^7}{r}\right)^3\)[/tex], let's proceed step-by-step.
1. Simplify the inner expression:
[tex]\[
\frac{-5 r^2 s^7}{r}
\][/tex]
Notice that [tex]\(r^2\)[/tex] divided by [tex]\(r\)[/tex] simplifies as follows:
[tex]\[
\frac{r^2}{r} = r^{2-1} = r^1 = r
\][/tex]
Therefore, the expression simplifies to:
[tex]\[
-5 r s^7
\][/tex]
2. Raise the simplified inner expression to the power of 3:
[tex]\[
(-5 r s^7)^3
\][/tex]
3. Apply the power to each term within the parentheses:
[tex]\[
(-5)^3 \cdot (r)^3 \cdot (s^7)^3
\][/tex]
4. Calculate each part separately:
- [tex]\((-5)^3\)[/tex]:
[tex]\[
(-5) \times (-5) \times (-5) = -125
\][/tex]
- [tex]\((r)^3\)[/tex]:
[tex]\[
r^3
\][/tex]
- [tex]\((s^7)^3\)[/tex]:
[tex]\[
(s^7)^3 = s^{7 \times 3} = s^{21}
\][/tex]
5. Combine all the parts together:
[tex]\[
(-125) \cdot r^3 \cdot s^{21} = -125 r^3 s^{21}
\][/tex]
Therefore, the simplified expression is:
[tex]\[
-125 r^3 s^{21}
\][/tex]
Among the given options, the correct one is:
[tex]\[
-125 r^3 s^{21}
\][/tex]
