Order of operations with Integers

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Understanding the **order of operations** is crucial when evaluating mathematical expressions, especially when integers (whole numbers that can be positive, negative, or zero) are involved. The **order of operations** dictates the sequence in which operations should be performed to obtain the correct result.

A common mnemonic to remember the order is **PEMDAS** or **BODMAS**:

- **PEMDAS**: 
  - **P**arentheses
  - **E**xponents
  - **M**ultiplication
  - **D**ivision
  - **A**ddition
  - **S**ubtraction

- **BODMAS**:
  - **B**rackets
  - **O**rders (i.e., exponents and roots)
  - **D**ivision
  - **M**ultiplication
  - **A**ddition
  - **S**ubtraction

**Key Points:**
1. **Parentheses/Brackets**: Always perform operations inside parentheses or brackets first.
2. **Exponents/Orders**: Solve exponents (powers and roots) next.
3. **Multiplication and Division**: These operations are of equal precedence and are performed from left to right.
4. **Addition and Subtraction**: These also have equal precedence and are performed from left to right.

### Applying Order of Operations with Integers

Let's walk through several examples to illustrate how to apply the order of operations when integers are involved.

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### **Example 1**

**Evaluate:**  
$$ 3 + 4 \times 2 $$

**Solution:**

1. **Multiplication first**:  
   $$4 \times 2 = 8$$

2. **Then addition**:  
   $$3 + 8 = 11$$

**Answer:**  
11

---

### **Example 2**

**Evaluate:**  
$$ ( -2 + 5 ) \times 3^2 $$

**Solution:**

1. **Parentheses first**:  
   $$ -2 + 5 = 3 $$

2. **Exponents next**:  
   $$3^2 = 9$$

3. **Multiplication**:  
   $$3 \times 9 = 27$$

**Answer:**  
27

---

### **Example 3**

**Evaluate:**  
$$ 6 \div 2 \times (1 + 2) $$

**Solution:**

1. **Parentheses first**:  
   $$1 + 2 = 3$$

2. **Division and Multiplication from left to right**:
   - $$6 \div 2 = 3$$
   - $$3 \times 3 = 9$$

**Answer:**  
9

---

### **Example 4**

**Evaluate:**  
$$ -3 + 4 \times (-2)^2 $$

**Solution:**

1. **Exponents first**:  
   $$(-2)^2 = 4$$  
   *(Note: The negative sign is included in the base, so squaring $$-2$$ gives $$4$$, not $$-4$$.)*

2. **Multiplication**:  
   $$4 \times 4 = 16$$

3. **Addition**:  
   $$-3 + 16 = 13$$

**Answer:**  
13

---

### **Example 5**

**Evaluate:**  
$$ - (3 + 4) \times 2 $$

**Solution:**

1. **Parentheses first**:  
   $$3 + 4 = 7$$

2. **Apply the negative sign**:  
   $$-7 \times 2 = -14$$

**Answer:**  
-14

---

### **Tips for Working with Integers and Order of Operations**

1. **Handle Parentheses Carefully:**  
   Operations inside parentheses or brackets should be handled first. If there's a negative sign before parentheses, remember it affects the entire expression inside.

2. **Exponents with Negative Bases:**  
   Pay attention to whether the negative sign is included in the base when dealing with exponents. For example, $$-3^2$$ is interpreted as $$-(3^2) = -9$$, whereas $$(-3)^2 = 9$$.

3. **Left-to-Right for Same Precedence Operations:**  
   When operations have the same precedence (like multiplication and division or addition and subtraction), perform them from left to right.

4. **Distributive Property:**  
   When a term outside the parentheses is multiplied by a sum inside, distribute the multiplication to each term inside. For example, $$2 \times (3 + 4) = 2 \times 3 + 2 \times 4 = 6 + 8 = 14$$.

5. **Double Check Negative Signs:**  
   Ensure that negative signs are correctly applied throughout the calculation. Misplacing a negative sign can lead to incorrect results.

---

### **Final Thought**

Mastering the order of operations with integers lays the foundation for solving more complex mathematical problems. Always follow PEMDAS/BODMAS systematically to ensure accuracy in your calculations.

If you have a specific problem or need further clarification, feel free to ask!
The order of operations is important for solving math problems with integers. Use PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) to perform operations in the correct sequence. Always handle parentheses first, then exponents, followed by multiplication and division from left to right, and finally addition and subtraction from left to right.

Order of operations with Integers

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