Simplifying Algebraic Expressions A Step-by-Step Guide

Jul 14, 2025 by ADMIN 55 views

$Simplifying the Expression (8-(8+3 \* 6))+-9 A Comprehensive Guide$

In the realm of mathematics, simplifying expressions is a fundamental skill. This article provides a comprehensive guide on how to simplify the expression $(8-(8+3 \* 6))+-9$ . We will break down each step, providing clear explanations and insights to enhance your understanding of the order of operations and algebraic simplification. This is crucial for anyone looking to build a solid foundation in algebra and beyond.

Understanding the Order of Operations

To accurately simplify any mathematical expression, it's essential to follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This order dictates the sequence in which operations should be performed to arrive at the correct answer. Let's delve into how PEMDAS applies to the expression at hand.

First, we address the parentheses. Inside the parentheses, we encounter $(8+3 \* 6)$ . According to PEMDAS, multiplication takes precedence over addition. Thus, we first multiply $3$ by $6$ , which yields $18$ . The expression inside the parentheses then becomes $(8+18)$ . Adding these numbers results in $26$ . Now, the original expression simplifies to $(8-26)+-9$ . This initial step is crucial, as it sets the stage for further simplification by resolving the innermost operations first. By adhering to PEMDAS, we ensure that the expression is simplified in a logical and mathematically sound manner. Mastering this principle is vital for tackling more complex algebraic problems and is a cornerstone of mathematical proficiency. Understanding the hierarchy of operations allows us to break down seemingly daunting expressions into manageable steps, leading to accurate and confident solutions. The correct application of PEMDAS not only simplifies calculations but also prevents common errors, making it an indispensable tool in mathematical problem-solving.

Step-by-Step Simplification of the Expression

Now, let's systematically simplify the expression $(8-(8+3 \* 6))+-9$ by following the order of operations. We've already established the importance of PEMDAS, and now we'll apply it step-by-step to reach the final simplified form. The journey of simplification begins with the innermost parentheses, working our way outwards while adhering to the correct sequence of operations. This meticulous approach not only ensures accuracy but also provides a clear path for others to follow, making the solution transparent and understandable. Each step is a building block, contributing to the overall simplification process and ultimately leading to a concise and manageable expression. This method is not just about arriving at the answer; it's about demonstrating a methodical and logical approach to problem-solving, a skill that is invaluable in mathematics and beyond.

1. Simplify Inside the Parentheses

We start by simplifying the expression inside the inner parentheses: $(8+3 \* 6)$ . As per PEMDAS, multiplication comes before addition. Therefore, we first perform the multiplication: $3 \* 6 = 18$ . The expression inside the parentheses now becomes $(8+18)$ . This step highlights the critical role of the order of operations in ensuring accurate simplification. Ignoring this order would lead to a completely different result, underscoring the necessity of adhering to PEMDAS. This initial simplification is a cornerstone of the entire process, laying the groundwork for subsequent steps and demonstrating the power of breaking down complex expressions into manageable components. By focusing on one operation at a time, we minimize the chances of errors and maintain a clear path towards the final solution. This approach is not only effective but also efficient, allowing us to tackle even the most intricate expressions with confidence.

2. Continue Simplifying Within Parentheses

Continuing with the parentheses, we now have $(8+18)$ . Adding these two numbers together, we get $26$ . So, the expression inside the parentheses simplifies to $26$ . This step is a straightforward application of addition, but it's essential to recognize its place within the broader context of the expression. The result, $26$ , is a crucial intermediate value that will be used in the next stage of simplification. This highlights the sequential nature of mathematical operations, where each step builds upon the previous one. By meticulously carrying out each calculation, we ensure the accuracy of the final answer. This phase of simplification demonstrates the importance of basic arithmetic skills in tackling more complex problems. A solid foundation in addition, subtraction, multiplication, and division is essential for navigating the intricacies of algebraic expressions and equations.

3. Address the Outer Parentheses

Now, the expression becomes $(8-26)+-9$ . We need to address the remaining parentheses. Subtracting $26$ from $8$ , we get $8-26 = -18$ . The expression now simplifies to $-18+-9$ . This step involves integer subtraction, a fundamental concept in mathematics. Understanding how to work with negative numbers is crucial for accurately simplifying expressions and solving equations. The result, $-18$ , is another key intermediate value that brings us closer to the final answer. By carefully performing this subtraction, we maintain the integrity of the expression and avoid common errors associated with negative numbers. This stage of simplification emphasizes the importance of paying close attention to signs and their impact on the outcome of calculations. A thorough understanding of integer operations is essential for success in algebra and beyond, enabling us to confidently navigate the complexities of mathematical expressions.

4. Final Simplification: Addition of Negative Numbers

Finally, we have $-18+-9$ . Adding these two negative numbers, we get $-18 + (-9) = -27$ . Therefore, the simplified expression is $-27$ . This final step is a clear demonstration of adding integers with the same sign, a basic yet crucial arithmetic operation. The result, $-27$ , is the culmination of all the previous steps, highlighting the importance of accuracy and attention to detail throughout the simplification process. This final calculation underscores the significance of understanding number properties and operations, which are fundamental to mathematical proficiency. By arriving at the simplified expression, we've successfully navigated the complexities of the original problem, showcasing the power of PEMDAS and step-by-step simplification techniques. This process is not only about finding the answer but also about developing a systematic approach to problem-solving, a skill that is valuable in various aspects of life.

Conclusion: The Simplified Expression

In conclusion, by meticulously following the order of operations (PEMDAS), we have successfully simplified the expression $(8-(8+3 \* 6))+-9$ to $-27$ . Each step, from addressing the innermost parentheses to performing the final addition of negative numbers, was crucial in arriving at the correct answer. This exercise underscores the importance of a systematic approach to mathematical problem-solving. The ability to break down complex expressions into manageable steps is a skill that is invaluable in mathematics and beyond. Mastering the order of operations is not just about getting the right answer; it's about developing a logical and methodical mindset that can be applied to a wide range of problems. The simplified expression, $-27$ , is a testament to the power of careful calculation and adherence to fundamental mathematical principles. This process of simplification not only enhances our understanding of algebraic expressions but also strengthens our problem-solving abilities, setting a solid foundation for future mathematical endeavors.