How To Evaluate Mathematical Expressions Step-by-Step

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Introduction

In this comprehensive guide, we will delve into the process of evaluating mathematical expressions. We'll dissect each expression step-by-step, emphasizing the order of operations and providing clear explanations to ensure a thorough understanding. Mastering the evaluation of mathematical expressions is fundamental to success in mathematics and various STEM fields. This article aims to provide a detailed walkthrough of how to solve expressions like: 48 + 6(5 × 6) × 13 + 162, (18 + 3) × 6 + (19 - 8) × 5, (12 + 3) + 3, (16 - 7) × 4, and 6 + 3 + 24 - 25 + 2. By breaking down each expression and meticulously applying the order of operations (PEMDAS/BODMAS), we will ensure clarity and accuracy in our solutions.

Understanding the Order of Operations

Before diving into the specific expressions, it's crucial to understand the order of operations. This set of rules ensures that mathematical expressions are evaluated consistently, leading to a single correct answer. The commonly used acronyms for the order of operations are PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) and BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction). Both acronyms represent the same hierarchy:

  1. Parentheses/Brackets: Evaluate expressions inside parentheses or brackets first.
  2. Exponents/Orders: Calculate exponents or orders (powers and square roots).
  3. Multiplication and Division: Perform multiplication and division from left to right.
  4. Addition and Subtraction: Perform addition and subtraction from left to right.

Adhering to this order is paramount to avoiding errors and achieving the correct results. Each operation plays a specific role in simplifying the expression, and tackling them in the right sequence ensures that the mathematical logic is maintained. For instance, performing multiplication before addition is crucial, as it directly impacts the outcome due to the multiplicative scaling effect. Similarly, correctly handling parentheses or brackets dictates which parts of the expression are treated as a single unit, thereby influencing the overall computation. Understanding PEMDAS/BODMAS is not just about memorizing an acronym; it's about grasping the underlying mathematical principles that govern how expressions are simplified and solved. By internalizing this order, individuals can confidently approach complex expressions, breaking them down into manageable steps, and ultimately arrive at accurate solutions. This foundation is essential for more advanced mathematical concepts and applications, making it a cornerstone of mathematical proficiency.

Evaluating 48 + 6(5 × 6) × 13 + 162

Let's start by evaluating the expression 48 + 6(5 × 6) × 13 + 162. This expression involves multiple operations, so we'll proceed step by step, adhering to the order of operations.

  1. Parentheses:
    • First, we evaluate the expression within the parentheses: 5 × 6 = 30.
    • The expression now becomes: 48 + 6(30) × 13 + 162.
  2. Multiplication:
    • Next, we perform the multiplications from left to right.
    • 6 × 30 = 180
    • The expression becomes: 48 + 180 × 13 + 162.
    • 180 × 13 = 2340
    • The expression becomes: 48 + 2340 + 162.
  3. Addition:
    • Finally, we perform the additions from left to right.
    • 48 + 2340 = 2388
    • The expression becomes: 2388 + 162.
    • 2388 + 162 = 2550

Therefore, the value of the expression 48 + 6(5 × 6) × 13 + 162 is 2550. This meticulous breakdown demonstrates the importance of adhering to PEMDAS/BODMAS. Initially, the expression seems complex due to the multitude of operations involved: addition, multiplication, and the presence of parentheses. However, by systematically addressing each part according to the established order, the expression is simplified step-by-step. Starting with the parentheses ensures that the product inside is treated as a single term before interacting with other elements of the expression. The subsequent multiplications are handled sequentially, transforming the expression into a series of additions. This stage-wise reduction not only makes the calculation more manageable but also minimizes the chances of error. The final addition consolidates all the intermediate results, leading to the ultimate solution. This process underscores the elegance of mathematical rules; they provide a structured approach to problem-solving, turning what appears to be a daunting task into a series of simpler, interconnected steps. Mastery of this methodical approach is invaluable for tackling more intricate mathematical problems in the future.

Evaluating (18 + 3) × 6 + (19 - 8) × 5

Now, let's evaluate the expression (18 + 3) × 6 + (19 - 8) × 5. This expression includes multiple sets of parentheses, multiplication, and addition. Again, we'll follow the order of operations.

  1. Parentheses:
    • Evaluate the expressions within the parentheses first.
    • (18 + 3) = 21
    • (19 - 8) = 11
    • The expression now becomes: 21 × 6 + 11 × 5
  2. Multiplication:
    • Perform the multiplications from left to right.
    • 21 × 6 = 126
    • 11 × 5 = 55
    • The expression becomes: 126 + 55
  3. Addition:
    • Finally, perform the addition.
    • 126 + 55 = 181

Therefore, the value of the expression (18 + 3) × 6 + (19 - 8) × 5 is 181. This example further illustrates the importance of handling parentheses before any other operations. The parentheses act as grouping symbols, ensuring that the enclosed operations are completed first, thereby correctly defining the structure of the expression. In this case, evaluating (18 + 3) and (19 - 8) at the outset simplifies the expression into a more manageable form involving only multiplication and addition. Once the parentheses are resolved, the multiplication operations are performed sequentially, maintaining the integrity of the mathematical relationships. The final step, addition, consolidates the intermediate products into a single, definitive answer. This methodical approach not only guarantees accuracy but also provides a clear pathway for understanding the flow of the calculation. Each step builds upon the previous one, transforming the initial complex expression into a straightforward sum. By consistently applying the rules of PEMDAS/BODMAS, individuals can confidently navigate through such expressions, ensuring that they arrive at the correct solution every time. This ability to dissect and solve complex expressions is crucial for advancing in mathematics and related fields.

Evaluating (12 + 3) + 3 and (16 - 7) × 4

Next, let's evaluate the expressions (12 + 3) + 3 and (16 - 7) × 4. These expressions are simpler but still require careful application of the order of operations.

Evaluating (12 + 3) + 3

  1. Parentheses:
    • Evaluate the expression within the parentheses:
    • (12 + 3) = 15
    • The expression becomes: 15 + 3
  2. Addition:
    • Perform the addition.
    • 15 + 3 = 18

Therefore, the value of the expression (12 + 3) + 3 is 18. The simplicity of this expression underscores a fundamental principle: even in seemingly straightforward calculations, adherence to the order of operations is essential. The presence of parentheses dictates that the addition within them must be performed first, which then simplifies the expression to a basic addition problem. This clear, step-by-step approach minimizes the chance of error and reinforces the importance of following established mathematical conventions. The initial grouping operation not only provides a structural cue but also ensures that the calculation proceeds logically, building upon intermediate results to reach the final answer. This attention to detail, even in basic arithmetic, lays the groundwork for tackling more complex mathematical challenges in the future. By mastering these foundational principles, individuals develop a consistent and reliable approach to problem-solving, fostering confidence in their mathematical abilities.

Evaluating (16 - 7) × 4

  1. Parentheses:
    • Evaluate the expression within the parentheses:
    • (16 - 7) = 9
    • The expression becomes: 9 × 4
  2. Multiplication:
    • Perform the multiplication.
    • 9 × 4 = 36

Therefore, the value of the expression (16 - 7) × 4 is 36. This calculation, similar to the previous one, highlights the crucial role of parentheses in determining the order of operations. The subtraction operation enclosed within the parentheses takes precedence, transforming the expression into a straightforward multiplication problem. This sequential simplification not only clarifies the computational path but also prevents potential misinterpretations that could arise if the operations were performed in the incorrect order. The result of the subtraction directly influences the subsequent multiplication, underscoring the interconnectedness of mathematical operations. This emphasis on structural precedence is a cornerstone of mathematical accuracy, enabling individuals to break down complex problems into manageable steps. By consistently applying these principles, one can confidently approach various mathematical challenges, ensuring that each operation is performed in its appropriate sequence, leading to correct and reliable solutions.

Evaluating 6 + 3 + 24 - 25 + 2

Finally, let's evaluate the expression 6 + 3 + 24 - 25 + 2. This expression involves only addition and subtraction, which are performed from left to right.

  1. Addition and Subtraction:
    • Perform the operations from left to right.
    • 6 + 3 = 9
    • The expression becomes: 9 + 24 - 25 + 2
    • 9 + 24 = 33
    • The expression becomes: 33 - 25 + 2
    • 33 - 25 = 8
    • The expression becomes: 8 + 2
    • 8 + 2 = 10

Therefore, the value of the expression 6 + 3 + 24 - 25 + 2 is 10. This final example showcases the simplicity and consistency of the order of operations when dealing with addition and subtraction. The rule of performing these operations from left to right is paramount to maintaining accuracy. Each step builds upon the previous one, with the result of each operation immediately influencing the subsequent calculation. This sequential approach ensures that the cumulative effect of each addition and subtraction is properly accounted for, leading to a correct final answer. The absence of parentheses or other higher-order operations simplifies the process, making it a clear demonstration of how basic arithmetic principles, when applied methodically, can efficiently solve mathematical problems. This understanding is crucial for building confidence in one's mathematical abilities and for tackling more complex expressions that may involve a combination of different operations.

Conclusion

In conclusion, we have thoroughly evaluated several mathematical expressions, emphasizing the importance of the order of operations (PEMDAS/BODMAS). By consistently applying these rules, we can accurately simplify and solve complex expressions. Mastering these skills is essential for further studies in mathematics and related fields. The key takeaway from this comprehensive guide is the indispensable nature of PEMDAS/BODMAS in the world of mathematics. We've meticulously dissected expressions like 48 + 6(5 × 6) × 13 + 162, (18 + 3) × 6 + (19 - 8) × 5, (12 + 3) + 3, (16 - 7) × 4, and 6 + 3 + 24 - 25 + 2, illustrating how each step, guided by the order of operations, contributes to the final solution. The presence of parentheses, exponents, multiplication, division, addition, and subtraction necessitates a structured approach to avoid computational errors. Understanding that parentheses and brackets take precedence ensures that grouped operations are treated as single entities before interacting with other elements of the expression. Exponents, representing powers and roots, define another layer of complexity that must be addressed before multiplication and division. The rule of performing multiplication and division from left to right, and likewise for addition and subtraction, further refines the order of calculations, ensuring consistency and accuracy. This step-by-step methodology not only provides a reliable means of solving mathematical problems but also fosters a deeper appreciation for the logic and structure inherent in mathematics. The ability to break down complex problems into simpler, manageable steps is a valuable skill that transcends mathematics, finding applications in various aspects of life. Ultimately, the mastery of the order of operations equips individuals with a powerful tool for problem-solving, enabling them to confidently tackle mathematical challenges and build a solid foundation for further learning.