https://file.aiquickdraw.com/mathbot/file/17281701625764cevk0ya.jpg

Let's solve each of the three problems step by step.

### Problem 1: 
\[
3 \frac{1}{3} - 4 \frac{2}{3} \div \frac{7}{9} + 5 \frac{1}{4} \cdot 2 \frac{2}{7}
\]

#### Step 1: Convert mixed numbers to improper fractions.

- \( 3 \frac{1}{3} = \frac{10}{3} \)
- \( 4 \frac{2}{3} = \frac{14}{3} \)
- \( 5 \frac{1}{4} = \frac{21}{4} \)
- \( 2 \frac{2}{7} = \frac{16}{7} \)

#### Step 2: Perform the operations.

1. \( \frac{14}{3} \div \frac{7}{9} = \frac{14}{3} \times \frac{9}{7} = \frac{126}{21} = 6 \)
2. Multiply \( \frac{21}{4} \times \frac{16}{7} = \frac{336}{28} = 12 \)

#### Step 3: Substitute the results and finish the expression.

\[
\frac{10}{3} - 6 + 12 = \frac{10}{3} + 6 = \frac{10}{3} + \frac{18}{3} = \frac{28}{3}
\]

Final answer for Problem 1:
\[
\frac{28}{3} \text{ or } 9 \frac{1}{3}
\]

---

### Problem 2:
\[
4 \cdot \left[ 3 \frac{1}{5} + 3 \frac{6}{11} \cdot \left( 7 \frac{1}{3} - 5 \frac{1}{2} \right) \right] + 8 \frac{5}{3}
\]

#### Step 1: Convert mixed numbers to improper fractions.

- \( 3 \frac{1}{5} = \frac{16}{5} \)
- \( 3 \frac{6}{11} = \frac{39}{11} \)
- \( 7 \frac{1}{3} = \frac{22}{3} \)
- \( 5 \frac{1}{2} = \frac{11}{2} \)
- \( 8 \frac{5}{3} = \frac{29}{3} \)

#### Step 2: Simplify the inner expression.

1. \( 7 \frac{1}{3} - 5 \frac{1}{2} = \frac{22}{3} - \frac{11}{2} = \frac{44}{6} - \frac{33}{6} = \frac{11}{6} \)
2. Multiply \( \frac{39}{11} \times \frac{11}{6} = \frac{429}{66} = \frac{13}{2} \)

#### Step 3: Continue solving.

\[
3 \frac{1}{5} + \frac{13}{2} = \frac{16}{5} + \frac{13}{2} = \frac{32}{10} + \frac{65}{10} = \frac{97}{10}
\]

#### Step 4: Multiply by 4 and add the final term.

\[
4 \cdot \frac{97}{10} = \frac{388}{10} = 38.8
\]
\[
38.8 + \frac{29}{3} = \frac{116.4}{3} + \frac{29}{3} = \frac{349.2}{30} \text{ approximately}
\]

---

### Problem 3:
\[
\frac{9^9 - 3(5 - 2) \div 9 - 1}{10 - 8 : [2(1 - 2)]}
\]

#### Step 1: Solve the inner expressions.

1. \( 5 - 2 = 3 \), so \( 3(5 - 2) = 9 \).
2. Now the expression becomes:
\[
\frac{9^9 - 9 \div 9 - 1}{10 - 8 : [2(1 - 2)]}
\]
3. \( 9 \div 9 = 1 \), so the numerator becomes:
\[
9^9 - 1 - 1
\]

#### Step 2: Simplify the denominator.

1. \( 1 - 2 = -1 \), so \( 2(-1) = -2 \), and \( 8 \div -2 = -4 \).
2. Therefore, the denominator is \( 10 - (-4) = 14 \).

#### Step 3: Put it all together.

The expression simplifies to:
\[
\frac{9^9 - 2}{14}
\]

Given the large number \( 9^9 \), this final result is quite large. Let me know if you'd like further clarification or details!

---

### Follow-up Questions:

1. How do you simplify mixed fractions in expressions?
2. What are the rules for dividing fractions and how did they apply in these problems?
3. How do you handle nested parentheses in complex mathematical expressions?
4. How does exponentiation like \( 9^9 \) affect the final results?
5. What steps would you take to evaluate very large powers such as \( 9^9 \)?

#### Tip: Always break down complex expressions step by step to avoid confusion and make calculations manageable!

1) 3 1/3 - 4 2/3 ÷ 7/9 + 5 1/4 ⋅ 2 2/7 = ?
2) 4 ⋅ [3 1/5 + 3 6/11 ⋅ (7 1/3 - 5 1/2)] + 8 5/3 = ?
3) (9 - 3(5 - 2):9 - 1) ÷ (10 - 8:[2(1 - 2)]) = ?

Learn to solve complex mixed fraction problems involving division and multiplication of fractions, with step-by-step solutions. This lesson also covers the order of operations with nested parentheses, suitable for students in Grades 6-9.

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