Answer :
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)} $[/tex]
Step 1: Start with the given rational expressions:
[tex]$ \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]
