Order of Operations Practice Worksheet #1 with problems and answer key.

Problem: Solve the given order of operations problems and explain the solution.

The worksheet provided involves solving expressions using the order of operations, often remembered by the acronym PEMDAS:
1. Parentheses
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.

---

Problem 1: \( 3 \times (2 \times 4^3) \div 4 \)

1. Parentheses: Evaluate the expression inside the parentheses first.
\[
2 \times 4^3
\]
- Calculate the exponent: \( 4^3 = 64 \).
- Multiply: \( 2 \times 64 = 128 \).

So, the expression becomes:
\[
3 \times 128 \div 4
\]

2. Multiplication and Division: Perform multiplication and division from left to right.
- Multiply: \( 3 \times 128 = 384 \).
- Divide: \( 384 \div 4 = 96 \).

Final Answer: \( 96 \)

---

Problem 2: \( (4^3 + 2 - 1) \)

1. Parentheses: Evaluate the expression inside the parentheses.
- Calculate the exponent: \( 4^3 = 64 \).
- Add and subtract: \( 64 + 2 - 1 = 65 \).

Final Answer: \( 65 \)

---

Problem 3: \( (5 \times 3) \times 1 + 5 \)

1. Parentheses: Evaluate the expression inside the parentheses.
- Multiply: \( 5 \times 3 = 15 \).

So, the expression becomes:
\[
15 \times 1 + 5
\]

2. Multiplication: Perform the multiplication.
- Multiply: \( 15 \times 1 = 15 \).

So, the expression becomes:
\[
15 + 5
\]

3. Addition: Perform the addition.
- Add: \( 15 + 5 = 20 \).

Final Answer: \( 20 \)

---

Problem 4: \( (7^2 - 2^3 - 6) \)

1. Parentheses: Evaluate the expression inside the parentheses.
- Calculate the exponents: \( 7^2 = 49 \) and \( 2^3 = 8 \).
- Subtract: \( 49 - 8 = 41 \).
- Subtract again: \( 41 - 6 = 35 \).

Final Answer: \( 35 \)

---

Problem 5: \( (5^3 + 7) \times 2 \)

1. Parentheses: Evaluate the expression inside the parentheses.
- Calculate the exponent: \( 5^3 = 125 \).
- Add: \( 125 + 7 = 132 \).

So, the expression becomes:
\[
132 \times 2
\]

2. Multiplication: Perform the multiplication.
- Multiply: \( 132 \times 2 = 264 \).

Final Answer: \( 264 \)

---

Problem 6: \( 4 - (9 + 2^2 \div 2) \)

1. Parentheses: Evaluate the expression inside the parentheses.
- Calculate the exponent: \( 2^2 = 4 \).
- Divide: \( 4 \div 2 = 2 \).
- Add: \( 9 + 2 = 11 \).

So, the expression becomes:
\[
4 - 11
\]

2. Subtraction: Perform the subtraction.
- Subtract: \( 4 - 11 = -7 \).

Final Answer: \( -7 \)

---

Problem 7: \( 6 - (9 + 8^2 \times 1^3) + 5 \)

1. Parentheses: Evaluate the expression inside the parentheses.
- Calculate the exponents: \( 8^2 = 64 \) and \( 1^3 = 1 \).
- Multiply: \( 64 \times 1 = 64 \).
- Add: \( 9 + 64 = 73 \).

So, the expression becomes:
\[
6 - 73 + 5
\]

2. Subtraction and Addition: Perform the operations from left to right.
- Subtract: \( 6 - 73 = -67 \).
- Add: \( -67 + 5 = -62 \).

Final Answer: \( -62 \)

---

Problem 8: \( (2 \div 4 \times 8) \)

1. Parentheses: Evaluate the expression inside the parentheses.
- Divide: \( 2 \div 4 = 0.5 \).
- Multiply: \( 0.5 \times 8 = 4 \).

Final Answer: \( 4 \)

---

Problem 9: \( 8 - (3 + 4^3) \times 5 \)

1. Parentheses: Evaluate the expression inside the parentheses.
- Calculate the exponent: \( 4^3 = 64 \).
- Add: \( 3 + 64 = 67 \).

So, the expression becomes:
\[
8 - 67 \times 5
\]

2. Multiplication: Perform the multiplication.
- Multiply: \( 67 \times 5 = 335 \).

So, the expression becomes:
\[
8 - 335
\]

3. Subtraction: Perform the subtraction.
- Subtract: \( 8 - 335 = -327 \).

Final Answer: \( -327 \)

---

Problem 10: \( 5 \times (2^3 - 8) \times 5 \)

1. Parentheses: Evaluate the expression inside the parentheses.
- Calculate the exponent: \( 2^3 = 8 \).
- Subtract: \( 8 - 8 = 0 \).

So, the expression becomes:
\[
5 \times 0 \times 5
\]

2. Multiplication: Perform the multiplication.
- Multiply: \( 5 \times 0 = 0 \).
- Multiply again: \( 0 \times 5 = 0 \).

Final Answer: \( 0 \)

---

Problem 11: \( (9 \times 9 + 5) \)

1. Parentheses: Evaluate the expression inside the parentheses.
- Multiply: \( 9 \times 9 = 81 \).
- Add: \( 81 + 5 = 86 \).

Final Answer: \( 86 \)

---

Problem 12: \( (1 + 4 - 4) \)

1. Parentheses: Evaluate the expression inside the parentheses.
- Add: \( 1 + 4 = 5 \).
- Subtract: \( 5 - 4 = 1 \).

Final Answer: \( 1 \)

---

Problem 13: \( 5 \times (4 \div 1^2 + 8) \)

1. Parentheses: Evaluate the expression inside the parentheses.
- Calculate the exponent: \( 1^2 = 1 \).
- Divide: \( 4 \div 1 = 4 \).
- Add: \( 4 + 8 = 12 \).

So, the expression becomes:
\[
5 \times 12
\]

2. Multiplication: Perform the multiplication.
- Multiply: \( 5 \times 12 = 60 \).

Final Answer: \( 60 \)

---

Problem 14: \( (5 - 8^2 + 6 - 1) \)

1. Parentheses: Evaluate the expression inside the parentheses.
- Calculate the exponent: \( 8^2 = 64 \).
- Subtract: \( 5 - 64 = -59 \).
- Add: \( -59 + 6 = -53 \).
- Subtract: \( -53 - 1 = -54 \).

Final Answer: \( -54 \)

---

Problem 15: \( 2^2 \div (6 \div 9) - 5 \)

1. Parentheses: Evaluate the expression inside the parentheses.
- Divide: \( 6 \div 9 = \frac{2}{3} \).

So, the expression becomes:
\[
2^2 \div \frac{2}{3} - 5
\]

2. Exponent: Calculate the exponent.
- \( 2^2 = 4 \).

So, the expression becomes:
\[
4 \div \frac{2}{3} - 5
\]

3. Division: Divide by a fraction by multiplying by its reciprocal.
- \( 4 \div \frac{2}{3} = 4 \times \frac{3}{2} = 6 \).

So, the expression becomes:
\[
6 - 5
\]

4. Subtraction: Perform the subtraction.
- Subtract: \( 6 - 5 = 1 \).

Final Answer: \( 1 \)

---

Problem 16: \( (3 + 1^2 + 4) \)

1. Parentheses: Evaluate the expression inside the parentheses.
- Calculate the exponent: \( 1^2 = 1 \).
- Add: \( 3 + 1 = 4 \).
- Add again: \( 4 + 4 = 8 \).

Final Answer: \( 8 \)

---

Problem 17: \( 1^3 - (2 + 3 + 7) \times 5 \)

1. Parentheses: Evaluate the expression inside the parentheses.
- Add: \( 2 + 3 + 7 = 12 \).

So, the expression becomes:
\[
1^3 - 12 \times 5
\]

2. Exponent: Calculate the exponent.
- \( 1^3 = 1 \).

So, the expression becomes:
\[
1 - 12 \times 5
\]

3. Multiplication: Perform the multiplication.
- Multiply: \( 12 \times 5 = 60 \).

So, the expression becomes:
\[
1 - 60
\]

4. Subtraction: Perform the subtraction.
- Subtract: \( 1 - 60 = -59 \).

Final Answer: \( -59 \)

---

Problem 18: \( 3 \times (2^3 + 5) + 2 \)

1. Parentheses: Evaluate the expression inside the parentheses.
- Calculate the exponent: \( 2^3 = 8 \).
- Add: \( 8 + 5 = 13 \).

So, the expression becomes:
\[
3 \times 13 + 2
\]

2. Multiplication: Perform the multiplication.
- Multiply: \( 3 \times 13 = 39 \).

So, the expression becomes:
\[
39 + 2
\]

3. Addition: Perform the addition.
- Add: \( 39 + 2 = 41 \).

Final Answer: \( 41 \)

---

Problem 19: \( 9 \times (2^3 \div 4 \times 5) \)

1. Parentheses: Evaluate the expression inside the parentheses.
- Calculate the exponent: \( 2^3 = 8 \).
- Divide: \( 8 \div 4 = 2 \).
- Multiply: \( 2 \times 5 = 10 \).

So, the expression becomes:
\[
9 \times 10
\]

2. Multiplication: Perform the multiplication.
- Multiply: \( 9 \times 10 = 90 \).

Final Answer: \( 90 \)

---

Problem 20: \( (8 + 7 + 2 - 9) \)

1. Parentheses: Evaluate the expression inside the parentheses.
- Add: \( 8 + 7 = 15 \).
- Add again: \( 15 + 2 = 17 \).
- Subtract: \( 17 - 9 = 8 \).

Final Answer: \( 8 \)

---

Final Answers

\[
\boxed{
\begin{aligned}
1. & \ 96 \\
2. & \ 65 \\
3. & \ 20 \\
4. & \ 35 \\
5. & \ 264 \\
6. & \ -7 \\
7. & \ -62 \\
8. & \ 4 \\
9. & \ -327 \\
10. & \ 0 \\
11. & \ 86 \\
12. & \ 1 \\
13. & \ 60 \\
14. & \ -54 \\
15. & \ 1 \\
16. & \ 8 \\
17. & \ -59 \\
18. & \ 41 \\
19. & \ 90 \\
20. & \ 8 \\
\end{aligned}
}
\]