Simplify the expression:
[tex]\[
\left(\frac{-4 r^2 s^3}{r}\right)^3
\][/tex]

A. [tex]\(64 r^{12} s^9\)[/tex]

B. [tex]\(-64 r^{12} s^9\)[/tex]

C. [tex]\(-4 r^{12} s^9\)[/tex]

D. [tex]\(-64 r^4 s^6\)[/tex]



Answer :

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]