Let's multiply the given rational expressions step by step.
Step 1: Start with the given rational expressions:
[tex]$
\frac{r^{10} s}{r^2-r-42} \cdot \frac{5 r^2-37 r+14}{r s^7}
$[/tex]
Step 2: Multiply the numerators together:
[tex]$
r^{10} s \cdot (5 r^2 - 37 r + 14)
$[/tex]
We get:
[tex]$
r^{10} s (5 r^2 - 37 r + 14)
$[/tex]
Step 3: Multiply the denominators together:
[tex]$
(r^2 - r - 42) \cdot (r s^7)
$[/tex]
We get:
[tex]$
r s^7 (r^2 - r - 42)
$[/tex]
Putting it all together, we have:
[tex]$
\frac{r^{10} s (5 r^2 - 37 r + 14)}{r s^7 (r^2 - r - 42)}
$[/tex]
Step 4: Simplify the expression by canceling the common factors.
First, let's cancel out the common variable \( s \):
[tex]$
\frac{r^{10}}{s^6} \cdot \frac{5 r^2 - 37 r + 14}{r^2 - r - 42}
$[/tex]
The fraction now simplifies to:
[tex]$
\frac{r^{10} (5 r^2 - 37 r + 14)}{r s^6 (r^2 - r - 42)}
$[/tex]
Next, simplify by the common factor \( r \):
[tex]$
\frac{r^{9} (5 r^2 - 37 r + 14)}{s^6 (r^2 - r - 42)}
$[/tex]
After simplification, we obtain:
[tex]$
\frac{r^{9} (5 r - 2)}{s^6 (r + 6)}
$[/tex]
So, the final simplified form of the expression is:
[tex]$
\frac{r^{9} (5 r - 2)}{s^6 (r + 6)}
$[/tex]
