Simplifying The Expression (-6+8)-(-15)÷3×{-8-[-2+(8+2)÷(-3-2)]+(-4+8-5)

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In the realm of mathematics, complex expressions often appear daunting at first glance. However, by systematically breaking them down and applying the correct order of operations, we can unravel their intricacies and arrive at the solution. This article delves into the step-by-step simplification of the mathematical expression [(-6+8)-(-15)÷3]×{-8-[-2+(8+2)÷(-3-2)]+(-4+8-5)]​, providing a comprehensive explanation of each operation performed. Understanding the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), is crucial for accurate mathematical calculations. This principle dictates the sequence in which operations should be carried out to ensure a consistent and correct result. Let's embark on this mathematical journey, meticulously dissecting the expression and revealing its final value. Grasping the order of operations is not just about following rules; it's about developing a logical approach to problem-solving. It's a skill that extends beyond mathematics, influencing how we tackle complex tasks in various aspects of life. By mastering this concept, we empower ourselves to break down challenges into manageable steps, leading to clarity and effective solutions. As we proceed with simplifying the expression, we'll emphasize the importance of PEMDAS at each stage, demonstrating its practical application in achieving the correct answer. This process will not only enhance our understanding of mathematical operations but also cultivate a systematic mindset applicable to diverse situations.

Step-by-Step Simplification

1. Simplifying within the Parentheses

Our initial focus lies on simplifying the expressions within the parentheses. This aligns with the first step in the PEMDAS order of operations. We begin by addressing the innermost parentheses, working our way outwards to ensure accuracy.

The expression contains several sets of parentheses, each requiring individual attention. Starting with the innermost group, we encounter expressions such as (-6+8), (-15)÷3, (8+2), (-3-2), and (-4+8-5). These seemingly small components form the building blocks of the entire expression, and their correct simplification is paramount to the overall solution.

Let's break down each of these inner parentheses:

  • (-6+8): This involves the addition of two integers with opposite signs. Subtracting the smaller absolute value from the larger and assigning the sign of the larger number yields a result of 2.
  • (-15)÷3: Here, we perform division between a negative integer and a positive integer. The result will be negative, and 15 divided by 3 is 5, so the outcome is -5.
  • (8+2): This is a simple addition of two positive integers, resulting in 10.
  • (-3-2): This represents the subtraction of 2 from -3, which is equivalent to adding -2 to -3. The result is -5.
  • (-4+8-5): This involves a combination of addition and subtraction. First, we add -4 and 8, which gives us 4. Then, we subtract 5 from 4, resulting in -1.

By meticulously simplifying these inner parentheses, we pave the way for further calculations. Each step taken contributes to a more manageable form of the expression, bringing us closer to the final answer. This methodical approach underscores the significance of attention to detail in mathematics, where even minor errors can propagate and affect the final result.

2. Addressing Division within the Brackets

Having simplified the innermost parentheses, we now shift our attention to division, another key operation within the PEMDAS hierarchy. The expression contains a division operation within the brackets, specifically (-15)÷3 and (8+2)÷(-3-2), which we've already partially simplified in the previous step. Now, let's delve deeper into these divisions.

As we noted earlier, (-15)÷3 results in -5. This is a straightforward division operation that we've already addressed. However, the other division, (8+2)÷(-3-2), requires a closer look. We've already established that (8+2) equals 10 and (-3-2) equals -5. Therefore, the division now becomes 10÷(-5).

Dividing 10 by -5 yields a result of -2. This step is crucial as it further simplifies the expression within the curly braces, bringing us closer to isolating the main operations. The careful execution of division operations is essential, as it directly impacts the subsequent calculations and the final outcome.

This stage underscores the interconnectedness of mathematical operations. The simplification of parentheses paved the way for division, and the result of division will, in turn, influence the outcome of addition and subtraction. This sequential approach highlights the importance of understanding the order of operations and applying it consistently throughout the problem-solving process. By meticulously addressing each division operation, we maintain accuracy and ensure that the final result is mathematically sound.

3. Tackling the Brackets and Braces

With the division operations addressed, we now focus on simplifying the expressions within the brackets and braces. This involves a combination of addition and subtraction, adhering to the PEMDAS order of operations. The goal is to further condense the expression, making it more manageable for the final multiplication step.

The expression contains both square brackets [] and curly braces {}, each housing a series of operations. We'll tackle them systematically, starting with the innermost brackets and working outwards. This approach ensures that we maintain the correct order of operations and avoid potential errors.

Let's begin with the square brackets: [-2+(8+2)÷(-3-2)]. We've already simplified (8+2)÷(-3-2) to -2 in the previous step. Now, we add this result to -2: -2 + (-2) = -4. This simplifies the entire expression within the square brackets to -4.

Now, we move on to the curly braces: **-8-[-2+(8+2)÷(-3-2)]+(-4+8-5)]}**. We've established that the expression within the square brackets simplifies to -4, and we previously found that (-4+8-5) equals -1. Substituting these values into the curly braces, we get {-8 - (-4) + (-1). Remember that subtracting a negative number is the same as adding its positive counterpart. Therefore, -8 - (-4) becomes -8 + 4, which equals -4. Now, we add -1 to -4: -4 + (-1) = -5. So, the entire expression within the curly braces simplifies to -5.

By meticulously simplifying the expressions within the brackets and braces, we've significantly reduced the complexity of the overall expression. This step-by-step approach highlights the importance of breaking down a large problem into smaller, more manageable parts. It also demonstrates how the results of previous calculations contribute to subsequent steps, ultimately leading to the final solution.

4. Final Multiplication

Having meticulously simplified the expressions within the parentheses, division, brackets, and braces, we now arrive at the final operation: multiplication. This is the culmination of our step-by-step simplification process, bringing us to the solution of the original mathematical expression.

The expression has been reduced to [2 - (-5)] × {-5}. Let's address the remaining operations within the brackets. Subtracting a negative number is equivalent to adding its positive counterpart. Therefore, 2 - (-5) becomes 2 + 5, which equals 7.

Now, we have the simplified expression: 7 × (-5). Multiplying a positive number by a negative number results in a negative number. The product of 7 and 5 is 35. Therefore, 7 × (-5) equals -35.

Thus, the final solution to the mathematical expression [(-6+8)-(-15)÷3]×{-8-[-2+(8+2)÷(-3-2)]+(-4+8-5)]}​ is -35. This result is the culmination of a series of carefully executed operations, each guided by the principles of PEMDAS and a methodical approach to problem-solving.

The successful simplification of this complex expression underscores the power of breaking down challenges into manageable steps. It also reinforces the importance of understanding the order of operations and applying it consistently. This methodical approach is not only valuable in mathematics but also applicable to various aspects of life, where complex tasks can be effectively tackled by dissecting them into smaller, more digestible components.

Conclusion

In conclusion, simplifying complex mathematical expressions requires a systematic approach and a thorough understanding of the order of operations, commonly known as PEMDAS. By meticulously dissecting the expression [(-6+8)-(-15)÷3]×{-8-[-2+(8+2)÷(-3-2)]+(-4+8-5)]​, we have demonstrated how to unravel its intricacies and arrive at the solution. The process involved simplifying parentheses, addressing division, tackling brackets and braces, and finally, performing the multiplication.

Each step in the simplification process was crucial, building upon the previous one and contributing to the overall clarity of the expression. The order of operations served as our guiding principle, ensuring that we performed the calculations in the correct sequence and avoided potential errors. This methodical approach not only led us to the correct answer but also highlighted the importance of precision and attention to detail in mathematics.

The final result, -35, is a testament to the power of systematic problem-solving. By breaking down the complex expression into smaller, more manageable parts, we were able to navigate the various operations and arrive at a definitive solution. This approach is not only applicable to mathematical problems but also to various challenges in life, where complex tasks can be effectively tackled by dissecting them into smaller, more digestible components.

Furthermore, the process of simplifying this expression underscores the interconnectedness of mathematical concepts. Each operation, from parentheses to multiplication, plays a vital role in the overall solution. Understanding these relationships and applying them consistently is key to mastering mathematical problem-solving. As we conclude this exploration, we encourage readers to embrace a systematic approach to mathematical challenges, remembering that even the most daunting expressions can be unravelled with patience, precision, and a solid understanding of the fundamental principles.