Mastering Order Of Operations A Step By Step Guide

In the realm of mathematics, the order of operations stands as a fundamental principle, a cornerstone that ensures consistent and accurate calculations. Often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) or BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction), this set of rules dictates the sequence in which mathematical operations must be performed to arrive at the correct solution. Ignoring this order can lead to drastically different and incorrect results. This article delves deep into the order of operations, providing a comprehensive understanding of each step and demonstrating its application through a series of solved exercises. By mastering this essential concept, you'll build a solid foundation for tackling more complex mathematical problems with confidence.

Understanding the Order of Operations

The order of operations is not an arbitrary set of rules; it's a logical framework designed to maintain consistency and clarity in mathematical calculations. Imagine a world without it – different people could interpret the same expression in various ways, leading to a chaotic landscape of conflicting answers. PEMDAS/BODMAS provides the structure we need, ensuring everyone arrives at the same solution when working with a given mathematical expression.

Let's break down each component of PEMDAS/BODMAS:

1. Parentheses/Brackets: These symbols act as containers, indicating that the operations within them should be performed first. This includes all types of brackets: parentheses (), square brackets [], and curly braces {}. When dealing with nested brackets (one set inside another), work from the innermost set outwards.
2. Exponents/Orders: Exponents (also known as powers or orders) represent repeated multiplication. For example, 3² (3 squared) means 3 multiplied by itself (3 * 3 = 9). This step takes precedence over multiplication, division, addition, and subtraction.
3. Multiplication and Division: These operations hold equal priority and are performed from left to right. If an expression contains both multiplication and division, you work across the expression in the order they appear.
4. Addition and Subtraction: Similar to multiplication and division, addition and subtraction have equal priority and are performed from left to right. The order in which they appear in the expression dictates the sequence of operations.

It's crucial to remember that these steps are not merely a suggestion; they are a strict protocol. Deviating from this order will almost certainly lead to an incorrect answer. To solidify your understanding, let's apply these principles to a series of exercises.

Solving Exercises Step-by-Step

Now, let's put our knowledge of the order of operations into practice by tackling the exercises provided. We'll break down each problem step-by-step, highlighting the application of PEMDAS/BODMAS at every stage.

Exercise a) $[(2-5)+6]+8-11+(4-3)=$

This exercise involves multiple levels of grouping and various arithmetic operations. Let's approach it systematically:

1. Innermost Parentheses: Begin by simplifying the expressions within the innermost parentheses: (2 - 5) = -3 and (4 - 3) = 1. The expression now becomes: [-3 + 6] + 8 - 11 + 1
2. Square Brackets: Next, simplify the expression within the square brackets: -3 + 6 = 3. The expression is now: 3 + 8 - 11 + 1
3. Addition and Subtraction (left to right): Perform the remaining operations from left to right: 3 + 8 = 11, then 11 - 11 = 0, and finally 0 + 1 = 1

Therefore, the solution to exercise a) is 1.

Exercise b) $\sqrt[3]{-15+7}-(-2)${1-3^2+(-1)3}$+(-1)^0=$

This exercise introduces cube roots, exponents, and negative numbers, adding complexity to the calculation. Let's break it down:

1. Cube Root and Innermost Parentheses: First, simplify within the cube root: -15 + 7 = -8. The cube root of -8 is -2 (since -2 * -2 * -2 = -8). Simultaneously, simplify the exponent: (-1)^0=1. The expression now becomes: -2 - (-2)${1-3^2+(-1)^3}$+1.
2. Exponents within Brackets: Next, deal with the exponents inside the square brackets: 3^2 = 9 and (-1)^3 = -1. The expression within the brackets becomes: 1 - 9 + (-1)
3. Simplifying Brackets: Perform the operations within the brackets from left to right: 1 - 9 = -8, then -8 + (-1) = -9. The expression now becomes: -2 - (-2) * (-9) + 1
4. Multiplication: Perform the multiplication: (-2) * (-9) = 18. The expression is now: -2 - 18 + 1
5. Addition and Subtraction (left to right): Perform the remaining operations from left to right: -2 - 18 = -20, then -20 + 1 = -19

Therefore, the solution to exercise b) is -19.

Exercise c) $\sqrt{-18} \cdot \sqrt{-2}+(-1)^5-2\left[-1+(-3)^2\right]=$

This exercise introduces square roots of negative numbers, which involve imaginary numbers, and further reinforces the order of operations.

1. Square Roots and Exponents: First, simplify the square roots and the exponent. Remember that the square root of a negative number involves the imaginary unit 'i', where i² = -1. So, √-18 = √(18 * -1) = √18 * √-1 = 3√2 * i and √-2 = √(2 * -1) = √2 * √-1 = √2 * i. Also, (-1)^5 = -1. The expression becomes: (3√2 * i) * (√2 * i) - 1 - 2[-1 + (-3)²]
2. Exponents within Brackets: Next, simplify the exponent inside the brackets: (-3)² = 9. The expression within the brackets becomes: -1 + 9
3. Simplifying Brackets: Perform the operation within the brackets: -1 + 9 = 8. The expression now becomes: (3√2 * i) * (√2 * i) - 1 - 2 * 8
4. Multiplication: Perform the multiplications: (3√2 * i) * (√2 * i) = 3 * 2 * i² = 6 * (-1) = -6 and 2 * 8 = 16. The expression is now: -6 - 1 - 16
5. Addition and Subtraction (left to right): Perform the remaining operations from left to right: -6 - 1 = -7, then -7 - 16 = -23

Therefore, the solution to exercise c) is -23.

Exercise d) $9-4 \cdot 5+6-76+8 \cdot 2=$

This exercise primarily involves multiplication, addition, and subtraction. Let's apply PEMDAS/BODMAS:

1. Multiplication: Perform the multiplications from left to right: 4 * 5 = 20 and 8 * 2 = 16. The expression becomes: 9 - 20 + 6 - 76 + 16
2. Addition and Subtraction (left to right): Perform the remaining operations from left to right: 9 - 20 = -11, then -11 + 6 = -5, then -5 - 76 = -81, and finally -81 + 16 = -65

Therefore, the solution to exercise d) is -65.

Key Takeaways and Common Pitfalls

Mastering the order of operations is not just about memorizing PEMDAS/BODMAS; it's about understanding the underlying logic and applying it consistently. Here are some key takeaways and common pitfalls to avoid:

• Always follow PEMDAS/BODMAS: This is the golden rule. Deviating from the order will lead to incorrect answers.
• Work from left to right for operations of equal priority: Multiplication and division, as well as addition and subtraction, should be performed in the order they appear from left to right.
• Pay close attention to signs: Negative signs can easily be overlooked, leading to errors. Be meticulous in tracking them throughout the calculation.
• Don't skip steps: Break down complex expressions into smaller, manageable steps. This reduces the chance of making mistakes.
• Practice regularly: The more you practice, the more comfortable you'll become with applying the order of operations.

Common pitfalls include:

• Adding before multiplying: This is a frequent error. Remember that multiplication and division take precedence over addition and subtraction.
• Ignoring negative signs: Failing to account for negative signs can drastically alter the result.
• Misinterpreting exponents: Ensure you understand what an exponent represents (repeated multiplication) and apply it correctly.

Conclusion

The order of operations is a cornerstone of mathematical accuracy. By understanding and consistently applying PEMDAS/BODMAS, you can confidently tackle a wide range of mathematical problems. This article has provided a comprehensive guide, from the fundamental principles to step-by-step solutions of complex exercises. Remember to practice regularly, pay close attention to detail, and avoid common pitfalls. With dedication and a solid grasp of the order of operations, you'll unlock a new level of mathematical proficiency. Mastering this skill not only enhances your problem-solving abilities but also lays the groundwork for success in more advanced mathematical concepts. Embrace the challenge, and you'll find that mathematics becomes less daunting and more rewarding.

Let's tackle some math problems that require us to follow the correct order of operations, often remembered by the acronym PEMDAS or BODMAS. This order is crucial for getting the right answer. PEMDAS stands for Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). BODMAS represents Brackets, Orders, Division and Multiplication (from left to right), Addition and Subtraction (from left to right). Understanding and applying this order is key to solving mathematical expressions accurately. This article provides a detailed walkthrough of each step, ensuring clarity and understanding of the principles involved. The importance of adhering to the correct sequence cannot be overstated; deviations can lead to significantly different and incorrect results. This in-depth exploration serves to solidify your comprehension and enhance your ability to confidently tackle complex mathematical problems.

Exercise Solutions

a) $[(2-5)+6]+8-11+(4-3)=$

This problem involves multiple parentheses and brackets, so we'll work from the innermost outwards, strictly adhering to PEMDAS/BODMAS. This methodical approach is essential for simplifying complex expressions and arriving at the correct solution.

1. First, we solve the innermost parentheses: (2 - 5) = -3 and (4 - 3) = 1. Replacing these values, our expression becomes: [-3 + 6] + 8 - 11 + 1. This step demonstrates the priority of parentheses in the order of operations. Accurate simplification at this stage is crucial for the subsequent steps. Ignoring this step or performing it incorrectly will undoubtedly lead to an incorrect final answer. Mastering this initial stage of simplifying expressions is fundamental to tackling more complex problems with multiple operations and nested groupings.
2. Next, we deal with the square brackets: [-3 + 6] = 3. Our expression simplifies further to: 3 + 8 - 11 + 1. The proper handling of brackets and parentheses is paramount in adhering to the correct order of operations. This step emphasizes the hierarchical nature of these groupings, where inner groupings must be resolved before outer ones. By simplifying the expression within the brackets, we're progressively reducing the complexity and moving closer to the final solution. Consistent application of this principle is essential for accurate mathematical calculations.
3. Now, we perform addition and subtraction from left to right: 3 + 8 = 11, then 11 - 11 = 0, and finally 0 + 1 = 1. This final sequence of operations highlights the equal precedence of addition and subtraction, where they are executed in the order they appear from left to right. This meticulous approach ensures that the operations are performed in the correct sequence, preventing potential errors. The expression is now fully simplified, leading us to the final answer. This step-by-step execution underscores the importance of methodical calculations in achieving accurate results in mathematics.

Therefore, the answer to a) is 1. This final result showcases the effectiveness of following the order of operations diligently. Each step, meticulously executed, contributes to the accurate simplification of the expression and ultimately leads to the correct solution. This process highlights the importance of precision and attention to detail in mathematical problem-solving.

b) $\sqrt[3]{-15+7}-(-2)${1-3^2+(-1)3}$+(-1)^0=$

This problem incorporates cube roots, exponents, and various operations within brackets, demanding a careful application of PEMDAS/BODMAS. The presence of these different elements necessitates a structured approach to ensure accurate simplification.

1. First, we simplify the cube root and the exponent outside the brackets: $\sqrt[3]{-15+7} = \sqrt[3]{-8} = -2$ and $(-1)^0 = 1$. Simultaneously, we simplify the exponent: (-1)^0=1. The expression now becomes: -2 - (-2)${1-3^2+(-1)^3}$+1. Handling the cube root and the exponent at the outset streamlines the subsequent steps by simplifying these complex operations first. The accurate computation of these initial elements is crucial for the overall correctness of the solution. This proactive simplification contributes to a clearer and more manageable expression to work with.
2. Next, we simplify the exponents within the brackets: $3^2 = 9$ and $(-1)^3 = -1$. The expression inside the brackets becomes: 1 - 9 + (-1). Prioritizing the exponents within the brackets aligns perfectly with the PEMDAS/BODMAS order. By addressing these operations first, we pave the way for subsequent simplification within the brackets. Accurate evaluation of exponents is essential, as any error here will propagate through the rest of the calculation. This step underscores the importance of attending to each operation in the correct sequence.
3. We simplify the expression within the square brackets: 1 - 9 + (-1) = -8 - 1 = -9. The expression now reads: -2 - (-2) * (-9) + 1. The meticulous simplification within the brackets is critical to obtaining the correct intermediate result. By systematically performing the operations in the correct order, we ensure the accuracy of the value within the brackets. This step highlights the significance of paying attention to signs and order to avoid any potential miscalculations.
4. Now, we perform the multiplication: -(-2) * (-9) = -18. Our expression is now: -2 - 18 + 1. The correct handling of multiplication with negative numbers is vital at this stage. This step demonstrates the importance of applying the rules of signs to ensure accurate multiplication. Any error in this step will affect the final result, emphasizing the need for careful calculation. Accurate multiplication is crucial for maintaining the integrity of the mathematical expression.
5. Finally, we perform addition and subtraction from left to right: -2 - 18 + 1 = -20 + 1 = -19. The sequential execution of addition and subtraction from left to right ensures the correct final result. This step underscores the equal precedence of these operations and the importance of their order. By carefully performing each operation, we arrive at the final solution. This methodical approach to the final stage highlights the significance of precision in mathematical calculations.

Therefore, the answer to b) is -19. This final answer demonstrates the successful application of PEMDAS/BODMAS in solving a complex mathematical problem. Each step, meticulously executed, contributes to the accurate simplification of the expression and the correct solution. The journey from the initial expression to the final answer highlights the power of a structured approach in mathematics.

c) $\sqrt{-18} \cdot \sqrt{-2}+(-1)^5-2\left[-1+(-3)^2\right]=$

This problem presents the challenge of square roots of negative numbers, which involve imaginary units, along with exponents and brackets, requiring a strong understanding of order of operations. The presence of complex numbers adds an extra layer of complexity, necessitating careful application of mathematical rules.

1. First, we handle the square roots of negative numbers and the exponent: $\sqrt{-18} = 3i\sqrt{2}$, $\sqrt{-2} = i\sqrt{2}$, and $(-1)^5 = -1$. Substituting these values, our expression becomes: $3i\sqrt{2} \cdot i\sqrt{2} - 1 - 2[-1 + (-3)^2]$. Simplifying the square roots and the exponent at the outset makes the subsequent steps more manageable. This initial simplification involves working with imaginary units, requiring careful attention to their properties. This step underscores the importance of understanding complex numbers and their role in mathematical expressions.
2. Next, we simplify the exponent within the brackets: $(-3)^2 = 9$. The expression inside the brackets becomes: -1 + 9. Prioritizing the exponent within the brackets is consistent with the PEMDAS/BODMAS order. By evaluating the exponent first, we pave the way for further simplification within the brackets. Accurate computation of the exponent is crucial for the overall correctness of the solution.
3. Now, we simplify the expression within the brackets: -1 + 9 = 8. Our expression simplifies to: $3i\sqrt{2} \cdot i\sqrt{2} - 1 - 2 * 8$. The meticulous simplification within the brackets is critical to obtaining the correct intermediate result. This step demonstrates the importance of performing the operations inside the brackets in the correct order. This accurate simplification ensures that the value within the brackets is correctly represented.
4. We perform the multiplication: $3i\sqrt{2} \cdot i\sqrt{2} = 3 * i^2 * 2 = 3 * (-1) * 2 = -6$ and 2 * 8 = 16. Our expression now reads: -6 - 1 - 16. The correct handling of multiplication involving imaginary units is vital at this stage. This step demonstrates the application of the properties of imaginary numbers, where i² = -1. Any error in this multiplication will affect the final result. Accurate multiplication is crucial for maintaining the integrity of the mathematical expression.
5. Finally, we perform addition and subtraction from left to right: -6 - 1 - 16 = -7 - 16 = -23. The sequential execution of addition and subtraction from left to right ensures the correct final result. This step underscores the equal precedence of these operations and the importance of their order. By carefully performing each operation, we arrive at the final solution. This methodical approach to the final stage highlights the significance of precision in mathematical calculations.

Therefore, the answer to c) is -23. This final answer showcases the successful application of PEMDAS/BODMAS in solving a complex mathematical problem involving imaginary units. Each step, meticulously executed, contributes to the accurate simplification of the expression and the correct solution. The journey from the initial expression to the final answer highlights the power of a structured approach in mathematics, even when dealing with complex numbers.

d) $9-4 \cdot 5+6-76+8 \cdot 2=$

This problem involves a mix of multiplication, addition, and subtraction, emphasizing the importance of adhering to the correct order of operations. This exercise provides a clear demonstration of how PEMDAS/BODMAS dictates the sequence of calculations.

1. First, we perform the multiplication operations: 4 * 5 = 20 and 8 * 2 = 16. Replacing these values, the expression becomes: 9 - 20 + 6 - 76 + 16. Prioritizing multiplication in accordance with PEMDAS/BODMAS is crucial for obtaining the correct result. By performing the multiplications before addition and subtraction, we ensure that the mathematical integrity of the expression is maintained. Accurate multiplication is fundamental to the subsequent steps and the final answer.
2. Now, we perform addition and subtraction from left to right: 9 - 20 = -11, then -11 + 6 = -5, then -5 - 76 = -81, and finally -81 + 16 = -65. This series of operations highlights the equal precedence of addition and subtraction, which are performed in the order they appear from left to right. This meticulous, sequential approach prevents potential errors and ensures the correct application of the order of operations. Each operation builds upon the previous one, leading us progressively closer to the final solution.

Therefore, the answer to d) is -65. This final answer demonstrates the successful application of PEMDAS/BODMAS in solving a mathematical problem involving multiple operations. By adhering to the correct order, we have accurately simplified the expression and arrived at the correct solution. This process underscores the importance of a systematic approach in mathematics, where each step is carefully executed to ensure accuracy.

Conclusion

In conclusion, mastering the order of operations is paramount for success in mathematics. By consistently applying PEMDAS/BODMAS, we can confidently solve complex expressions and arrive at accurate solutions. The exercises presented here illustrate the importance of each step in the order of operations and the potential pitfalls of deviating from this order. Continued practice and a deep understanding of these principles will empower you to tackle a wide range of mathematical challenges with confidence. Remember, the order of operations is not just a set of rules; it's a logical framework that ensures consistency and accuracy in mathematical calculations. Embrace this framework, and you'll unlock a new level of mathematical proficiency.

The order of operations, a cornerstone of mathematics, ensures consistent and accurate results when solving mathematical expressions. Often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) or BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction), this set of rules dictates the sequence in which mathematical operations must be performed. Neglecting this order can lead to incorrect solutions, highlighting the importance of understanding and applying these principles. This article provides a detailed guide to solving mathematical exercises while respecting the order of operations, enhancing your understanding and problem-solving skills. This comprehensive guide aims to demystify the order of operations, providing clear explanations and practical examples. By mastering this fundamental concept, you'll build a solid foundation for tackling more advanced mathematical topics. The emphasis on step-by-step solutions ensures that every aspect of the order of operations is clearly understood and can be applied confidently.

Understanding the Principles of PEMDAS/BODMAS

The acronyms PEMDAS and BODMAS serve as mnemonic devices to remember the correct order of operations. Let's delve into each component:

• P/B - Parentheses/Brackets: Operations within parentheses or brackets are performed first. This includes all types of grouping symbols: parentheses (), square brackets [], and curly braces {}. When dealing with nested brackets, work from the innermost set outwards. This step underscores the priority of grouped operations, which must be simplified before any other operations are performed. Accurate handling of parentheses and brackets is essential for maintaining the mathematical integrity of the expression.
• E/O - Exponents/Orders: Exponents (also known as powers or orders) represent repeated multiplication. For example, 3² (3 squared) means 3 multiplied by itself (3 * 3 = 9). This step takes precedence over multiplication, division, addition, and subtraction. This highlights the hierarchical nature of exponents, which are evaluated before other arithmetic operations. Correct computation of exponents is vital for obtaining the correct solution.
• MD - Multiplication and Division: These operations hold equal priority and are performed from left to right. If an expression contains both multiplication and division, you work across the expression in the order they appear. The left-to-right rule for multiplication and division emphasizes the sequential nature of these operations. This ensures consistency and prevents ambiguity in calculations. Adhering to this rule is critical for accurate mathematical problem-solving.
• AS - Addition and Subtraction: Similar to multiplication and division, addition and subtraction have equal priority and are performed from left to right. The order in which they appear in the expression dictates the sequence of operations. The left-to-right rule also applies to addition and subtraction, reinforcing the importance of sequential calculations. This ensures that the operations are performed in the correct order, leading to the accurate final result. This consistent approach is key to mastering the order of operations.

It's crucial to remember that these steps are not merely suggestions; they are a strict protocol. Deviating from this order will almost certainly lead to an incorrect answer. To solidify your understanding, let's apply these principles to a series of exercises. The adherence to PEMDAS/BODMAS is not optional, but rather a fundamental requirement for mathematical accuracy. This strict adherence ensures that everyone arrives at the same solution when working with a given mathematical expression.

Solved Exercises with Detailed Explanations

Now, let's apply the principles of PEMDAS/BODMAS to solve the given exercises step-by-step. This practical application will reinforce your understanding and improve your problem-solving skills. The following exercises provide a comprehensive demonstration of how to apply the order of operations in various mathematical scenarios. Each step is carefully explained to ensure clarity and facilitate learning.

a) Solve: $[(2-5)+6]+8-11+(4-3)=$

This exercise involves multiple levels of grouping and various arithmetic operations. Let's approach it systematically:

1. Simplify Innermost Parentheses: Begin by simplifying the expressions within the innermost parentheses: (2 - 5) = -3 and (4 - 3) = 1. The expression now becomes: [-3 + 6] + 8 - 11 + 1. Addressing the innermost groupings first is a cornerstone of the order of operations. This methodical approach allows us to gradually simplify the expression and prevent potential errors. Accurate simplification at this stage is crucial for the subsequent steps.
2. Simplify Square Brackets: Next, simplify the expression within the square brackets: -3 + 6 = 3. The expression is now: 3 + 8 - 11 + 1. Simplifying the square brackets follows directly from simplifying the innermost parentheses. This step-by-step reduction of the expression's complexity is essential for maintaining accuracy. The proper handling of brackets is paramount in adhering to the correct order of operations.
3. Perform Addition and Subtraction (left to right): Perform the remaining operations from left to right: 3 + 8 = 11, then 11 - 11 = 0, and finally 0 + 1 = 1. This final sequence of operations highlights the equal precedence of addition and subtraction, where they are executed in the order they appear from left to right. This meticulous approach ensures that the operations are performed in the correct sequence, preventing potential errors. The sequential execution of addition and subtraction is vital for achieving the accurate final result.

Therefore, the solution to exercise a) is 1. This final answer demonstrates the successful application of PEMDAS/BODMAS in solving the mathematical expression. Each step, meticulously executed, contributes to the accurate simplification and the correct solution. This process underscores the importance of a structured approach in mathematical problem-solving.

b) Solve: $\sqrt[3]{-15+7}-(-2)${1-3^2+(-1)3}$+(-1)^0=$

This exercise incorporates cube roots, exponents, and negative numbers, adding complexity to the calculation. Let's break it down step-by-step:

1. Simplify Cube Root and Exponent: First, simplify within the cube root: -15 + 7 = -8. The cube root of -8 is -2 (since -2 * -2 * -2 = -8). Simultaneously, simplify the exponent outside the brackets: (-1)^0 = 1. The expression now becomes: -2 - (-2)${1-3^2+(-1)^3}$ + 1. Addressing the cube root and exponent at the outset streamlines the subsequent steps by simplifying these complex operations first. The accurate computation of these initial elements is crucial for the overall correctness of the solution. This proactive simplification contributes to a clearer and more manageable expression to work with.
2. Simplify Exponents within Brackets: Next, deal with the exponents inside the square brackets: 3^2 = 9 and (-1)^3 = -1. The expression within the brackets becomes: 1 - 9 + (-1). Prioritizing the exponents within the brackets aligns perfectly with the PEMDAS/BODMAS order. By addressing these operations first, we pave the way for subsequent simplification within the brackets. Accurate evaluation of exponents is essential, as any error here will propagate through the rest of the calculation.
3. Simplify Expression within Brackets: Perform the operations within the brackets from left to right: 1 - 9 = -8, then -8 + (-1) = -9. The expression now becomes: -2 - (-2) * (-9) + 1. The meticulous simplification within the brackets is critical to obtaining the correct intermediate result. By systematically performing the operations in the correct order, we ensure the accuracy of the value within the brackets. This step highlights the significance of paying attention to signs and order to avoid any potential miscalculations.
4. Perform Multiplication: Perform the multiplication: (-2) * (-9) = 18. The expression is now: -2 - 18 + 1. The correct handling of multiplication with negative numbers is vital at this stage. This step demonstrates the importance of applying the rules of signs to ensure accurate multiplication. Any error in this step will affect the final result, emphasizing the need for careful calculation. Accurate multiplication is crucial for maintaining the integrity of the mathematical expression.
5. Perform Addition and Subtraction (left to right): Perform the remaining operations from left to right: -2 - 18 = -20, then -20 + 1 = -19. The sequential execution of addition and subtraction from left to right ensures the correct final result. This step underscores the equal precedence of these operations and the importance of their order. By carefully performing each operation, we arrive at the final solution. This methodical approach to the final stage highlights the significance of precision in mathematical calculations.

Therefore, the solution to exercise b) is -19. This final answer demonstrates the successful application of PEMDAS/BODMAS in solving a complex mathematical problem. Each step, meticulously executed, contributes to the accurate simplification of the expression and the correct solution. The journey from the initial expression to the final answer highlights the power of a structured approach in mathematics.

c) Solve: $\sqrt{-18} \cdot \sqrt{-2}+(-1)^5-2\left[-1+(-3)^2\right]=$

This exercise introduces square roots of negative numbers, which involve imaginary units, and further reinforces the order of operations. The presence of complex numbers adds an extra layer of complexity, necessitating careful application of mathematical rules.

1. Simplify Square Roots and Exponent: First, simplify the square roots and the exponent. Remember that the square root of a negative number involves the imaginary unit 'i', where i² = -1. So, √-18 = √(18 * -1) = √18 * √-1 = 3√2 * i and √-2 = √(2 * -1) = √2 * √-1 = √2 * i. Also, (-1)^5 = -1. The expression becomes: (3√2 * i) * (√2 * i) - 1 - 2[-1 + (-3)²]. Simplifying the square roots and the exponent at the outset makes the subsequent steps more manageable. This initial simplification involves working with imaginary units, requiring careful attention to their properties. This step underscores the importance of understanding complex numbers and their role in mathematical expressions.
2. Simplify Exponent within Brackets: Next, simplify the exponent inside the brackets: (-3)² = 9. The expression within the brackets becomes: -1 + 9. Prioritizing the exponent within the brackets is consistent with the PEMDAS/BODMAS order. By evaluating the exponent first, we pave the way for further simplification within the brackets. Accurate computation of the exponent is crucial for the overall correctness of the solution.
3. Simplify Expression within Brackets: Now, simplify the expression within the brackets: -1 + 9 = 8. Our expression simplifies to: (3√2 * i) * (√2 * i) - 1 - 2 * 8. The meticulous simplification within the brackets is critical to obtaining the correct intermediate result. This step demonstrates the importance of performing the operations inside the brackets in the correct order. This accurate simplification ensures that the value within the brackets is correctly represented.
4. Perform Multiplication: Perform the multiplications: (3√2 * i) * (√2 * i) = 3 * 2 * i² = 6 * (-1) = -6 and 2 * 8 = 16. Our expression now reads: -6 - 1 - 16. The correct handling of multiplication involving imaginary units is vital at this stage. This step demonstrates the application of the properties of imaginary numbers, where i² = -1. Any error in this multiplication will affect the final result. Accurate multiplication is crucial for maintaining the integrity of the mathematical expression.
5. Perform Addition and Subtraction (left to right): Finally, we perform addition and subtraction from left to right: -6 - 1 - 16 = -7 - 16 = -23. The sequential execution of addition and subtraction from left to right ensures the correct final result. This step underscores the equal precedence of these operations and the importance of their order. By carefully performing each operation, we arrive at the final solution. This methodical approach to the final stage highlights the significance of precision in mathematical calculations.

Therefore, the solution to exercise c) is -23. This final answer showcases the successful application of PEMDAS/BODMAS in solving a complex mathematical problem involving imaginary units. Each step, meticulously executed, contributes to the accurate simplification of the expression and the correct solution. The journey from the initial expression to the final answer highlights the power of a structured approach in mathematics, even when dealing with complex numbers.

d) Solve: $9-4 \cdot 5+6-76+8 \cdot 2=$

This exercise primarily involves multiplication, addition, and subtraction. Let's apply PEMDAS/BODMAS:

1. Perform Multiplication: Perform the multiplications from left to right: 4 * 5 = 20 and 8 * 2 = 16. The expression becomes: 9 - 20 + 6 - 76 + 16. Prioritizing multiplication in accordance with PEMDAS/BODMAS is crucial for obtaining the correct result. By performing the multiplications before addition and subtraction, we ensure that the mathematical integrity of the expression is maintained. Accurate multiplication is fundamental to the subsequent steps and the final answer.
2. Perform Addition and Subtraction (left to right): Perform the remaining operations from left to right: 9 - 20 = -11, then -11 + 6 = -5, then -5 - 76 = -81, and finally -81 + 16 = -65. This series of operations highlights the equal precedence of addition and subtraction, which are performed in the order they appear from left to right. This meticulous, sequential approach prevents potential errors and ensures the correct application of the order of operations. Each operation builds upon the previous one, leading us progressively closer to the final solution.

Therefore, the solution to exercise d) is -65. This final answer demonstrates the successful application of PEMDAS/BODMAS in solving a mathematical problem involving multiple operations. By adhering to the correct order, we have accurately simplified the expression and arrived at the correct solution. This process underscores the importance of a systematic approach in mathematics, where each step is carefully executed to ensure accuracy.

Common Mistakes and How to Avoid Them

Even with a strong understanding of PEMDAS/BODMAS, common mistakes can occur. Recognizing these errors and implementing strategies to avoid them is crucial for mathematical proficiency. Identifying common pitfalls is an essential aspect of mastering the order of operations. By understanding the types of errors that are frequently made, you can develop strategies to avoid them and improve your accuracy.

• Adding before Multiplying: One of the most frequent errors is performing addition before multiplication. Remember that multiplication and division take precedence over addition and subtraction. This error stems from a misunderstanding of the hierarchy within PEMDAS/BODMAS. Always ensure that multiplication and division are performed before addition and subtraction. Consistent application of this rule is essential for avoiding this common mistake.
• Ignoring Negative Signs: Failing to account for negative signs can drastically alter the result. Be meticulous in tracking them throughout the calculation. Negative signs can be easily overlooked, leading to incorrect answers. Pay close attention to the sign of each number and ensure that it is correctly incorporated into the calculations. This attention to detail is crucial for maintaining accuracy.
• Misinterpreting Exponents: Ensure you understand what an exponent represents (repeated multiplication) and apply it correctly. For instance, 3² means 3 * 3, not 3 * 2. A clear understanding of exponents is crucial for accurate calculations. Misinterpreting exponents can lead to significant errors in the final result. Regular practice can help solidify this understanding and prevent mistakes.
• Incorrectly Applying the Left-to-Right Rule: For operations of equal priority (multiplication and division, or addition and subtraction), perform them from left to right. Deviating from this rule can lead to errors. The left-to-right rule is essential for maintaining consistency when dealing with operations of equal precedence. Always perform these operations in the order they appear from left to right. This methodical approach ensures accuracy and avoids ambiguity.

To avoid these mistakes, it's helpful to write out each step clearly and double-check your work. Additionally, regular practice is key to solidifying your understanding and building confidence. Consistent application of PEMDAS/BODMAS, combined with careful attention to detail, will significantly reduce the likelihood of errors. Embrace a structured approach to problem-solving, and you'll find that mathematics becomes less daunting and more rewarding.

Conclusion

Mastering the order of operations is a fundamental skill in mathematics. By understanding and consistently applying PEMDAS/BODMAS, you can confidently tackle a wide range of mathematical problems. This article has provided a comprehensive guide, from the fundamental principles to step-by-step solutions of complex exercises, along with common mistakes to avoid. Remember to practice regularly, pay close attention to detail, and adopt a structured approach to problem-solving. The journey to mathematical proficiency begins with a solid understanding of the order of operations. By mastering this essential skill, you'll unlock new opportunities for learning and problem-solving. Embrace the challenge and the rewards that come with mathematical mastery.