Order of Operations Practice Worksheet for solving mathematical expressions.

To solve the problems in the "Order of Operations Practice" worksheet, we need to follow the order of operations, often remembered by the acronym PEMDAS:

1. Parentheses (and other grouping symbols like brackets)
2. Exponents
3. Multiplication and Division (from left to right)
4. Addition and Subtraction (from left to right)

Let's solve each problem step by step.

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Problem 1: \( 3 + [(9 \div 3) \cdot 2] + [(2 \cdot 6) \div 3] \)

1. Solve inside the parentheses:
- \( 9 \div 3 = 3 \)
- \( 2 \cdot 6 = 12 \)

2. Substitute back:
- \( 3 + [3 \cdot 2] + [12 \div 3] \)

3. Perform multiplication and division:
- \( 3 \cdot 2 = 6 \)
- \( 12 \div 3 = 4 \)

4. Substitute back:
- \( 3 + 6 + 4 \)

5. Perform addition:
- \( 3 + 6 = 9 \)
- \( 9 + 4 = 13 \)

Answer: \( 13 \)

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Problem 2: \( [(3 + 9) \div (3 \cdot 2)] + [(2 \cdot 6) \div 3] \)

1. Solve inside the parentheses:
- \( 3 + 9 = 12 \)
- \( 3 \cdot 2 = 6 \)
- \( 2 \cdot 6 = 12 \)

2. Substitute back:
- \( [12 \div 6] + [12 \div 3] \)

3. Perform division:
- \( 12 \div 6 = 2 \)
- \( 12 \div 3 = 4 \)

4. Substitute back:
- \( 2 + 4 \)

5. Perform addition:
- \( 2 + 4 = 6 \)

Answer: \( 6 \)

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Problem 3: \( (3 + 9) \div [(3 \cdot 2 + 2 \cdot 6) \div 3] \)

1. Solve inside the parentheses:
- \( 3 + 9 = 12 \)
- \( 3 \cdot 2 = 6 \)
- \( 2 \cdot 6 = 12 \)

2. Add inside the second set of parentheses:
- \( 6 + 12 = 18 \)

3. Substitute back:
- \( 12 \div [18 \div 3] \)

4. Perform division inside the brackets:
- \( 18 \div 3 = 6 \)

5. Substitute back:
- \( 12 \div 6 \)

6. Perform division:
- \( 12 \div 6 = 2 \)

Answer: \( 2 \)

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Problem 4: \( (3 + 9 \div 3 \cdot 2) + (2 \cdot 6 \div 3) \)

1. Solve inside the first set of parentheses:
- \( 9 \div 3 = 3 \)
- \( 3 \cdot 2 = 6 \)
- \( 3 + 6 = 9 \)

2. Solve inside the second set of parentheses:
- \( 2 \cdot 6 = 12 \)
- \( 12 \div 3 = 4 \)

3. Substitute back:
- \( 9 + 4 \)

4. Perform addition:
- \( 9 + 4 = 13 \)

Answer: \( 13 \)

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Problem 5: \( [(3 + 9) \div 3 \cdot 2 + 2] \cdot (6 \div 3) \)

1. Solve inside the first set of brackets:
- \( 3 + 9 = 12 \)
- \( 12 \div 3 = 4 \)
- \( 4 \cdot 2 = 8 \)
- \( 8 + 2 = 10 \)

2. Solve inside the second set of parentheses:
- \( 6 \div 3 = 2 \)

3. Substitute back:
- \( 10 \cdot 2 \)

4. Perform multiplication:
- \( 10 \cdot 2 = 20 \)

Answer: \( 20 \)

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Problem 6: \( [(3 + 9 \div 3) \cdot 2] + (2 \cdot 6 \div 3) \)

1. Solve inside the first set of brackets:
- \( 9 \div 3 = 3 \)
- \( 3 + 3 = 6 \)
- \( 6 \cdot 2 = 12 \)

2. Solve inside the second set of parentheses:
- \( 2 \cdot 6 = 12 \)
- \( 12 \div 3 = 4 \)

3. Substitute back:
- \( 12 + 4 \)

4. Perform addition:
- \( 12 + 4 = 16 \)

Answer: \( 16 \)

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Problem 7: \( [(3 + 9 \div (3 \cdot 2)) + 2] \cdot 6 \div 3 \)

1. Solve inside the innermost parentheses:
- \( 3 \cdot 2 = 6 \)
- \( 9 \div 6 = 1.5 \)

2. Solve inside the next set of parentheses:
- \( 3 + 1.5 = 4.5 \)
- \( 4.5 + 2 = 6.5 \)

3. Multiply:
- \( 6.5 \cdot 6 = 39 \)

4. Divide:
- \( 39 \div 3 = 13 \)

Answer: \( 13 \)

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Problem 8: \( 3 + (9 \div 3) \cdot (2 + 2 \cdot 6 \div 3) \)

1. Solve inside the first set of parentheses:
- \( 9 \div 3 = 3 \)

2. Solve inside the second set of parentheses:
- \( 2 \cdot 6 = 12 \)
- \( 12 \div 3 = 4 \)
- \( 2 + 4 = 6 \)

3. Multiply:
- \( 3 \cdot 6 = 18 \)

4. Add:
- \( 3 + 18 = 21 \)

Answer: \( 21 \)

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Problem 9: \( 3 + [((9 \div 3) \cdot (2 + 2)) \cdot (6 \div 3)] \)

1. Solve inside the innermost parentheses:
- \( 9 \div 3 = 3 \)
- \( 2 + 2 = 4 \)

2. Multiply:
- \( 3 \cdot 4 = 12 \)

3. Solve the division:
- \( 6 \div 3 = 2 \)

4. Multiply:
- \( 12 \cdot 2 = 24 \)

5. Add:
- \( 3 + 24 = 27 \)

Answer: \( 27 \)

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Problem 10: \( [(3 + 9) \div 3] \cdot 2 + [(2 \cdot 6) \div 3] \)

1. Solve inside the first set of brackets:
- \( 3 + 9 = 12 \)
- \( 12 \div 3 = 4 \)

2. Multiply:
- \( 4 \cdot 2 = 8 \)

3. Solve inside the second set of brackets:
- \( 2 \cdot 6 = 12 \)
- \( 12 \div 3 = 4 \)

4. Add:
- \( 8 + 4 = 12 \)

Answer: \( 12 \)

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Final Answers:

1. \( 13 \)
2. \( 6 \)
3. \( 2 \)
4. \( 13 \)
5. \( 20 \)
6. \( 16 \)
7. \( 13 \)
8. \( 21 \)
9. \( 27 \)
10. \( 12 \)

\[
\boxed{13, 6, 2, 13, 20, 16, 13, 21, 27, 12}
\]