Solving Numerical Expressions A, B, C, And D A Step-by-Step Guide
In the realm of mathematics, numerical expressions serve as fundamental building blocks for more complex equations and formulas. Mastering the simplification and evaluation of these expressions is crucial for students and professionals alike. This comprehensive guide delves into the step-by-step solutions of four distinct numerical expressions, labeled A, B, C, and D, providing a clear and concise understanding of the underlying mathematical principles involved.
Dissecting Expression A: A=(-5) . 5+15
Expression A presents a combination of multiplication and addition operations. To accurately solve this expression, we must adhere to the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This order dictates that we perform multiplication before addition. Beginning with the multiplication, we have (-5) . 5, which results in -25. Subsequently, we add 15 to this product: -25 + 15. The sum of these two numbers is -10. Therefore, the simplified value of expression A is -10. This process highlights the importance of understanding the order of operations and how negative numbers interact in mathematical calculations. The expression A serves as a foundational example, illustrating how basic arithmetic operations combine to form a numerical expression. By correctly applying the order of operations and managing the signs of the numbers involved, we can confidently arrive at the solution. In essence, expression A acts as a microcosm of more complex mathematical problems, providing a valuable learning opportunity for understanding the fundamentals of arithmetic and algebraic manipulations. The systematic approach employed here, starting with multiplication and progressing to addition, is a cornerstone of mathematical problem-solving and serves as a vital tool for anyone looking to improve their mathematical proficiency. Further practice with similar expressions will solidify this understanding and enhance the ability to tackle more challenging mathematical scenarios.
Deconstructing Expression B: B=9 . 3-18÷(-3)
Expression B introduces both multiplication, subtraction and division, demanding careful application of the order of operations. Following PEMDAS, we first address multiplication and division from left to right. The multiplication operation 9 . 3 yields 27. Next, we perform the division: 18 ÷ (-3). Dividing a positive number by a negative number results in a negative quotient, so 18 ÷ (-3) equals -6. Now, the expression is simplified to 27 - (-6). Subtracting a negative number is equivalent to adding its positive counterpart. Therefore, 27 - (-6) becomes 27 + 6. The sum of these two numbers is 33. Thus, the final value of expression B is 33. The solution to expression B highlights the importance of paying close attention to the signs of the numbers involved and how they interact during arithmetic operations. The presence of both subtraction and division, combined with a negative divisor, adds a layer of complexity that necessitates a thorough understanding of mathematical rules. This example serves as an excellent illustration of how the order of operations ensures consistency and accuracy in mathematical calculations. By systematically addressing each operation according to PEMDAS, we can break down complex expressions into manageable steps and arrive at the correct solution. The experience gained from solving expression B will be invaluable in tackling more advanced mathematical problems involving a variety of operations and numerical values. Furthermore, the emphasis on sign management reinforces the crucial role of precision in mathematical calculations, a skill that is essential for success in various mathematical disciplines.
Solving Expression C: C=2³+(36÷9-2)0
Expression C brings exponents and parentheses into the mix, further emphasizing the importance of the order of operations. According to PEMDAS, we start with parentheses and then exponents. Inside the parentheses, we have 36 ÷ 9 - 2. First, we perform the division: 36 ÷ 9 = 4. Then, we subtract 2: 4 - 2 = 2. So, the expression inside the parentheses simplifies to 2. Now, we address the exponent: 2³ (2 cubed) means 2 multiplied by itself three times, which is 2 . 2 . 2 = 8. The expression now becomes 8 + 2. Finally, we perform the addition: 8 + 2 = 10. Therefore, the value of expression C is 10. The introduction of exponents in expression C requires a solid grasp of exponential notation and how it fits within the order of operations. Cubing the number 2 (2³) signifies multiplying 2 by itself three times, a concept that is fundamental to understanding exponents and powers in mathematics. Moreover, the parentheses in expression C serve to group a set of operations, which must be performed before any operations outside the parentheses are considered. This underscores the critical role of parentheses in dictating the sequence of calculations and ensuring the correct evaluation of complex expressions. The process of solving expression C exemplifies how multiple mathematical concepts can come together in a single problem, demanding a comprehensive understanding of arithmetic operations, exponents, and the order of operations. By systematically addressing each element of the expression according to PEMDAS, we can navigate the complexities and arrive at the accurate solution. This experience is invaluable for building mathematical confidence and preparing for more challenging problems that involve a combination of mathematical concepts.
Unraveling Expression D: D=(-4) . (-6)÷[2 . (-7)+2]
Expression D presents a more intricate combination of operations, including multiplication, division, addition, and nested parentheses (brackets), requiring meticulous application of the order of operations. We begin by tackling the innermost parentheses first: 2 . (-7) = -14. Now, the expression within the brackets becomes -14 + 2, which equals -12. So, the brackets simplify to -12. Next, we address the multiplication: (-4) . (-6). Multiplying two negative numbers results in a positive product, so (-4) . (-6) = 24. The expression is now 24 ÷ (-12). Finally, we perform the division: 24 ÷ (-12). Dividing a positive number by a negative number results in a negative quotient, so 24 ÷ (-12) = -2. Therefore, the value of expression D is -2. Expression D stands as a testament to the importance of meticulous attention to detail when dealing with complex mathematical expressions. The presence of nested parentheses necessitates a systematic approach, starting with the innermost grouping and working outwards. This strategy ensures that the operations are performed in the correct sequence, preventing errors and leading to an accurate solution. Furthermore, the expression involves both positive and negative numbers, requiring a firm understanding of how these numbers interact during multiplication and division. The process of solving expression D exemplifies the power of breaking down a complex problem into smaller, more manageable steps. By systematically addressing each operation according to the order of operations and paying close attention to the signs of the numbers involved, we can navigate the intricacies and arrive at the correct answer. This experience is invaluable for building problem-solving skills and preparing for more challenging mathematical scenarios that involve a combination of operations and nested groupings.
In conclusion, by systematically applying the order of operations (PEMDAS) and carefully managing the signs of the numbers, we have successfully solved expressions A, B, C, and D. This exercise highlights the fundamental principles of arithmetic and provides a solid foundation for tackling more complex mathematical problems. Each expression presented unique challenges, from the basic combination of multiplication and addition in expression A to the intricate interplay of operations and nested parentheses in expression D. Through the step-by-step solutions provided, we have demonstrated how a methodical approach and a thorough understanding of mathematical rules can lead to accurate results. This knowledge is invaluable for students, professionals, and anyone seeking to enhance their mathematical proficiency. The ability to confidently solve numerical expressions is a cornerstone of mathematical literacy and a crucial skill for success in various fields. By mastering the principles illustrated in this guide, individuals can unlock their mathematical potential and approach complex problems with clarity and precision.