Let's go through a step-by-step approach to solve these problems. I'll begin with question 1 and then explain subsequent problems in similar detail.

Problem 1:

$\frac{7(m - 6)}{m - 6} \cdot \frac{5m(7m - 5)}{7(7m - 5)}$

Step 1: Simplify each fraction

• The first fraction can be simplified as: \frac{7(m - 6)}{m - 6} = 7 \quad \text{(since the (m - 6) cancels out)}
• The second fraction simplifies as: \frac{5m(7m - 5)}{7(7m - 5)} = \frac{5m}{7} \quad \text{(as the (7m - 5) cancels out)}

Step 2: Multiply the simplified fractions $7 \cdot \frac{5m}{7} = 5m$

So, the final answer to problem 1 is: $\boxed{5m}$

Would you like me to continue with problem 2, or do you have any specific questions so far?

Five Follow-Up Questions:

1. How can you handle division of rational expressions more efficiently?
2. What should you do when a factor cancels in both the numerator and denominator?
3. Why is it important to check for restrictions (e.g., $m \neq 6$) in rational expressions?
4. How does simplifying fractions before multiplying save time in these problems?
5. How would this process change if terms were added inside the numerator and denominator?

Tip:

Always cancel common factors before multiplying to simplify the expression.