Simplifying The Expression 43192A - (35 + S - 3) + (26 + 35 - 40) + 84 - 15

by ADMIN 76 views

Navigating the world of mathematics often involves deciphering complex expressions, and this exploration delves into the intricacies of simplifying the expression 43192A - (35 + s - 3) + (26 + 35 - 40) + 84 - 15. This article serves as a comprehensive guide, meticulously breaking down each step involved in solving this mathematical puzzle. Understanding the fundamental principles behind the order of operations is crucial for accurately simplifying such expressions. The journey begins by defining the core concepts and gradually progressing towards the final solution.

Understanding the Order of Operations

At the heart of simplifying mathematical expressions lies the order of operations, a set of rules that dictates the sequence in which operations must be performed. Often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), this order ensures consistency and accuracy in mathematical calculations. Let's dissect each component of PEMDAS:

  • Parentheses (or Brackets): Operations enclosed within parentheses or brackets take precedence over all others. This means any calculations inside parentheses must be completed before proceeding further. Parentheses act as a way to group terms and prioritize specific operations.
  • Exponents: Exponents represent repeated multiplication and are evaluated after parentheses. An exponent indicates how many times a base number is multiplied by itself. For example, in the expression 2^3, 2 is the base, and 3 is the exponent, meaning 2 is multiplied by itself three times (2 * 2 * 2 = 8).
  • Multiplication and Division: These operations hold equal priority and are performed from left to right. This means if multiplication and division appear in the same expression, you would perform the operation that comes first as you read from left to right.
  • Addition and Subtraction: Similar to multiplication and division, addition and subtraction also share equal priority and are executed from left to right. The operation that appears earlier in the expression (when reading from left to right) is performed first.

Mastering PEMDAS is essential for simplifying any mathematical expression correctly. Without adhering to this order, the outcome is highly likely to be inaccurate. Now, let's apply these principles to unravel the expression at hand.

Dissecting the Expression: 43192A - (35 + s - 3) + (26 + 35 - 40) + 84 - 15

The given expression 43192A - (35 + s - 3) + (26 + 35 - 40) + 84 - 15 presents a blend of arithmetic operations and an algebraic element ('s'). To effectively simplify this, we will meticulously follow PEMDAS, tackling the parentheses first and then proceeding with addition and subtraction from left to right. This methodical approach ensures accurate simplification.

Step 1: Simplifying Within Parentheses

The expression contains two sets of parentheses, each requiring simplification: (35 + s - 3) and (26 + 35 - 40). Let's begin with the first set.

  • (35 + s - 3): Within these parentheses, we encounter both addition and subtraction, along with the variable 's'. Combining the constants, 35 and -3, we get 32. Thus, the simplified form of this expression is 32 + s. The variable 's' remains as it is, as its value is not provided.
  • (26 + 35 - 40): This set of parentheses involves only numerical values. Performing the addition and subtraction from left to right: 26 + 35 equals 61, and then subtracting 40 from 61 yields 21. Therefore, (26 + 35 - 40) simplifies to 21.

Having simplified the expressions within the parentheses, our expression now looks like this: 43192A - (32 + s) + 21 + 84 - 15. The next step involves addressing the subtraction of the expression (32 + s). This requires distributing the negative sign, which we will discuss in the next section.

Step 2: Addressing the Negative Sign and Further Simplification

With the parentheses partially simplified, the expression stands as 43192A - (32 + s) + 21 + 84 - 15. The immediate next step involves handling the negative sign preceding the parentheses (32 + s). This is a crucial step as it affects the signs of the terms within the parentheses.

  • Distributing the Negative Sign: The negative sign in front of the parentheses implies multiplication by -1. Distributing this -1 across the terms inside the parentheses means multiplying both 32 and 's' by -1. This results in: -1 * 32 = -32 and -1 * s = -s. Consequently, -(32 + s) transforms into -32 - s. This step is essential to accurately remove the parentheses and proceed with further simplification.

Now, the expression is updated to 43192A - 32 - s + 21 + 84 - 15. All parentheses have been eliminated, paving the way for combining like terms. This involves grouping the constant numerical values and any terms containing the variable 's'.

Step 3: Combining Like Terms and Isolating Variables

Following the distribution of the negative sign, the expression is now 43192A - 32 - s + 21 + 84 - 15. The next crucial step involves combining like terms. This means grouping together the constant numerical values and any terms that include the variable 's'. This process simplifies the expression and brings us closer to the final solution.

  • Grouping Constants: The constant terms in the expression are -32, 21, 84, and -15. Let's combine them: -32 + 21 + 84 - 15. First, add -32 and 21, which gives -11. Then, add 84 to -11, resulting in 73. Finally, subtract 15 from 73, which equals 58. Therefore, the combined constant term is 58.
  • Identifying Variables: The expression includes the variable term '-s' and '43192A'. Since there are no other terms with 's' or 'A', these terms remain as they are. Note that '43192A' implies multiplication: 43192 multiplied by 'A'. Without a value for 'A', we cannot simplify this term further.

After combining like terms, the expression is now simplified to 43192A - s + 58. This form represents the most simplified version of the original expression, given the information available. The final step is to present this simplified expression and discuss its implications.

Final Simplified Expression and Conclusion

After meticulously applying the order of operations and combining like terms, the simplified form of the expression 43192A - (35 + s - 3) + (26 + 35 - 40) + 84 - 15 is determined to be 43192A - s + 58. This represents the most concise form of the original expression, given the absence of specific values for the variables 'A' and 's'.

The journey through this mathematical problem underscores the significance of adhering to the order of operations (PEMDAS). Each step, from simplifying within parentheses to distributing negative signs and combining like terms, plays a crucial role in arriving at the correct solution. Any deviation from this order can lead to errors and an incorrect final expression.

In this particular case, the presence of variables 'A' and 's' in the expression means that the final result is an algebraic expression rather than a single numerical value. To obtain a numerical solution, specific values for 'A' and 's' would need to be provided. For instance, if A = 1 and s = 10, the expression would evaluate to 43192(1) - 10 + 58 = 43240. This illustrates how the values of variables influence the final result.

In conclusion, simplifying mathematical expressions requires a blend of understanding fundamental principles, meticulous application of rules, and careful attention to detail. The expression 43192A - (35 + s - 3) + (26 + 35 - 40) + 84 - 15 serves as an excellent example of this, showcasing the power of PEMDAS and the importance of combining like terms. This exploration not only solves the problem at hand but also reinforces the core principles of mathematical simplification.