Simplify And Evaluate The Mathematical Expression

Let's dive into the world of numbers and tackle this intriguing mathematical expression: -1-4^2-rac{(-4)^2}{2}. Evaluating mathematical expressions might seem daunting at first, but with a clear understanding of the order of operations, it becomes a systematic and even enjoyable process. This expression, in particular, challenges us to apply several fundamental rules of arithmetic, including exponentiation and division. We'll break it down step-by-step, ensuring clarity and accuracy in our solution. Our goal is to simplify this expression to its most basic numerical form, demonstrating the power of mathematical precision.

Understanding the Order of Operations (PEMDAS/BODMAS)

Before we begin evaluating, it's crucial to recall the universally accepted order of operations, often remembered by the acronyms PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) or BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction). This hierarchy dictates the sequence in which mathematical operations must be performed to arrive at a unique and correct answer. Understanding the order of operations is the cornerstone of solving any mathematical problem involving multiple operations. Without it, different interpretations could lead to vastly different results, rendering mathematical communication ambiguous. In our expression, we have subtraction, exponentiation, and division, so applying PEMDAS correctly is paramount. We'll start with any operations inside parentheses (though none are strictly necessary here for evaluation order, the structure implies grouping), then move to exponents, followed by multiplication and division from left to right, and finally, addition and subtraction from left to right. This systematic approach ensures that we treat each part of the expression with the correct precedence, preventing common errors and leading us confidently towards the correct simplified value.

Step 1: Addressing Exponents

The first step in evaluating our expression, -1-4^2-rac{(-4)^2}{2}, involves dealing with the exponents. According to PEMDAS, exponents take precedence over multiplication, division, addition, and subtraction. We have two instances of exponentiation here: $4^2$ and $(-4)^2$. Let's evaluate each of these. First, $4^2$ means 4 multiplied by itself, which is $4 imes 4 = 16$. So, the expression now looks like -1 - 16 - rac{(-4)^2}{2}. Next, we have $(-4)^2$. This means $-4$ multiplied by itself. It's important to note that the parentheses around $-4$ indicate that the entire number, including the negative sign, is being squared. Therefore, $(-4) imes (-4) = 16$. Remember, a negative number multiplied by a negative number results in a positive number. So, the expression transforms into -1 - 16 - rac{16}{2}. By carefully handling these exponents, we've successfully simplified two key components of the original problem, bringing us closer to the final numerical answer.

Step 2: Performing Division

With the exponents resolved, the next operation to tackle according to PEMDAS is division. In our current expression, -1 - 16 - rac{16}{2}, we have a clear division operation: rac{16}{2}. This part of the expression asks us to divide 16 by 2. Performing this division, we get $16 ilde{A} ext{divide} 2 = 8$. Now, we can substitute this value back into our expression. The expression is now reduced to $-1 - 16 - 8$. This step is critical as it further simplifies the problem by resolving the fractional part, leaving us with only subtractions. The successful execution of this division brings us one step closer to our final answer, having systematically worked through the hierarchy of operations.

Step 3: Executing Subtraction

Finally, we arrive at the last set of operations: addition and subtraction. In our simplified expression, $-1 - 16 - 8$, we only have subtractions. When faced with multiple subtractions (or additions and subtractions), we perform them from left to right. First, we take the initial $-1$ and subtract $16$ from it. This gives us $-1 - 16 = -17$. Now, our expression is $-17 - 8$. The final step is to subtract $8$ from $-17$. Performing this final subtraction, we get $-17 - 8 = -25$. Thus, the evaluation of the entire expression -1-4^2-rac{(-4)^2}{2} leads us to the final answer of -25. This systematic approach, adhering strictly to the order of operations, ensures accuracy in simplifying complex mathematical expressions.

Conclusion

We have successfully navigated the evaluation of the mathematical expression -1-4^2-rac{(-4)^2}{2} by diligently following the order of operations (PEMDAS/BODMAS). We began by resolving the exponents ($4^2$ and $(-4)^2$), then proceeded to the division (rac{16}{2}), and finally concluded with the subtractions. Each step was crucial in simplifying the expression to its final value of -25. This detailed walkthrough highlights the importance of a methodical approach in mathematics, where understanding the rules and applying them consistently leads to accurate and reliable results. Whether you're a student grappling with algebra or simply refreshing your mathematical skills, remember that breaking down complex problems into smaller, manageable steps is key to success. For further exploration into mathematical principles and problem-solving techniques, consider visiting resources like Khan Academy which offers comprehensive guides and exercises on a wide range of mathematical topics, including order of operations and algebraic simplification.