Certainly! Let's simplify the expression step by step.
Given expression:
[tex]\[
\left(\frac{-8 r^6 s^3}{r}\right)^3
\][/tex]
### Step 1: Simplify inside the parentheses
First, simplify the fraction inside the parentheses:
[tex]\[
\frac{-8 r^6 s^3}{r}
\][/tex]
Since [tex]\( r^6 \)[/tex] divided by [tex]\( r \)[/tex] can be written as [tex]\( r^{6-1} \)[/tex]:
[tex]\[
\frac{-8 r^6 s^3}{r} = -8 r^{6-1} s^3 = -8 r^5 s^3
\][/tex]
### Step 2: Raise the simplified expression to the power of 3
Now, we need to raise [tex]\(-8 r^5 s^3\)[/tex] to the power of 3:
[tex]\[
\left(-8 r^5 s^3\right)^3
\][/tex]
### Step 3: Apply the power rule
When raising a product to a power, raise each factor to that power:
[tex]\[
\left(-8\right)^3 \left(r^5\right)^3 \left(s^3\right)^3
\][/tex]
### Step 4: Calculate each part
- [tex]\(\left(-8\right)^3 = -8 \times -8 \times -8 = -512\)[/tex]
- [tex]\(\left(r^5\right)^3 = r^{5 \times 3} = r^{15}\)[/tex]
- [tex]\(\left(s^3\right)^3 = s^{3 \times 3} = s^9\)[/tex]
### Step 5: Combine the results
Combining all these parts, the simplified expression is:
[tex]\[
-512 r^{15} s^9
\][/tex]
Thus, the simplified form of the given expression is:
[tex]\[
\boxed{-512 r^{15} s^9}
\][/tex]
