Let's simplify the given expression step by step:
1. Start with the original expression:
[tex]\[
\left(\frac{-4 r^2 s^3}{r}\right)^3
\][/tex]
2. Simplify the expression inside the parentheses first:
[tex]\[
\frac{-4 r^2 s^3}{r}
\][/tex]
3. Simplify the fraction by dividing [tex]\( r^2 \)[/tex] by [tex]\( r \)[/tex]:
[tex]\[
\frac{-4 r^2 s^3}{r} = -4 r^{2-1} s^3 = -4 r^1 s^3 = -4 r s^3
\][/tex]
4. Now take the above simplified expression and raise it to the power of 3:
[tex]\[
(-4 r s^3)^3
\][/tex]
5. Apply the power to each term separately:
- [tex]\((-4)^3\)[/tex]
- [tex]\(r^3\)[/tex]
- [tex]\((s^3)^3\)[/tex]
6. Compute each term's power:
- [tex]\((-4)^3 = -64\)[/tex]
- [tex]\(r^3\)[/tex]
- [tex]\((s^3)^3 = s^{3 \times 3} = s^9\)[/tex]
7. Combine all the terms:
[tex]\[
(-4 r s^3)^3 = -64 r^3 s^9
\][/tex]
So, the simplified expression is:
[tex]\[
-64 r^3 s^9
\][/tex]
Among the given choices, the correct answer is:
[tex]\[
-64 r^3 s^9
\][/tex]
