The expression in the image is:

$5 \frac{5}{7} \div 1 \frac{3}{5} \cdot 4 \frac{2}{3}$

To solve this, let's break it down into steps:

Step 1: Convert the Mixed Numbers to Improper Fractions

1. $5 \frac{5}{7} = \frac{5 \cdot 7 + 5}{7} = \frac{35 + 5}{7} = \frac{40}{7}$
2. $1 \frac{3}{5} = \frac{1 \cdot 5 + 3}{5} = \frac{5 + 3}{5} = \frac{8}{5}$
3. $4 \frac{2}{3} = \frac{4 \cdot 3 + 2}{3} = \frac{12 + 2}{3} = \frac{14}{3}$

Now the expression becomes:

$\frac{40}{7} \div \frac{8}{5} \cdot \frac{14}{3}$

Step 2: Perform the Division

Dividing by a fraction is the same as multiplying by its reciprocal:

$\frac{40}{7} \cdot \frac{5}{8} \cdot \frac{14}{3}$

Step 3: Simplify the Expression by Multiplying the Fractions

Multiply the numerators and denominators:

$\frac{40 \cdot 5 \cdot 14}{7 \cdot 8 \cdot 3}$

Step 4: Simplify by Canceling Common Factors

1. Simplify $\frac{40}{8} = 5$, so the expression becomes: $\frac{5 \cdot 5 \cdot 14}{7 \cdot 3} = \frac{25 \cdot 14}{21}$

2. Simplify $\frac{14}{7} = 2$, so now we have: $\frac{25 \cdot 2}{3} = \frac{50}{3}$

Final Answer

$\frac{50}{3} \quad \text{or as a mixed number: } 16 \frac{2}{3}$

Would you like a detailed breakdown of any part of the solution, or do you have any other questions?

Here are five related questions to extend this topic:

1. How do you multiply and divide mixed numbers directly without converting to improper fractions?
2. How do you simplify complex fractions in multi-step problems like this?
3. What strategies help in reducing fractions early in multi-step calculations?
4. How do you handle division and multiplication with negative fractions?
5. How do you convert a result in improper fraction form to a decimal?

Tip: Always simplify fractions step-by-step to make large multiplications easier and avoid calculation errors.