To simplify the given expression:
[tex]\[
\left(\frac{-4 r^2 s^3}{r}\right)^3
\][/tex]
Let's start by simplifying the fraction inside the parentheses:
1. [tex]\(\frac{-4 r^2 s^3}{r}\)[/tex]
Divide [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]
Now, our expression is:
[tex]\[
(-4 r s^3)^3
\][/tex]
Next, we apply the exponent to each component inside the parentheses:
2. [tex]\((-4 r s^3)^3\)[/tex] can be expanded as:
[tex]\[
(-4)^3 \cdot (r)^3 \cdot (s^3)^3
\][/tex]
3. Calculate each component separately:
- [tex]\((-4)^3 = -64\)[/tex]
- [tex]\(r^3 = r^3\)[/tex]
- [tex]\((s^3)^3 = s^{3 \cdot 3} = s^9\)[/tex]
Combine these results together:
4. Multiplying the components:
[tex]\[
(-4)^3 \cdot r^3 \cdot s^9 = -64 r^3 s^9
\][/tex]
Therefore, the simplified expression is:
[tex]\[
-64 r^3 s^9
\][/tex]
So, the correct answer is not one of the previously listed options, but instead should be [tex]\(-64 r^3 s^9\)[/tex], matching the simplified expression we derived.
