Order of operations with Integers

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. --- ### **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!