Evaluate The Expression 6(-3)^2 - 24 - (-8 - 4)

Introduction

In this mathematical exploration, we will delve into the process of evaluating the expression $6(-3)^2 - 24 - (-8 - 4)$. This expression combines several fundamental arithmetic operations, including exponentiation, multiplication, subtraction, and dealing with negative numbers. To arrive at the correct answer, we must adhere strictly to the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This meticulous approach ensures we navigate the expression systematically and accurately. Let's embark on this step-by-step journey to unravel the solution.

Understanding the order of operations is paramount in mathematics. It provides a consistent framework for evaluating expressions, ensuring that everyone arrives at the same answer. PEMDAS acts as our guiding principle. First, we tackle anything within parentheses. Next, we address exponents. Then comes multiplication and division (from left to right), followed by addition and subtraction (also from left to right). This hierarchical structure prevents ambiguity and guarantees uniformity in mathematical calculations. Without it, expressions could be interpreted in multiple ways, leading to a chaotic landscape of conflicting solutions. The order of operations is not merely a convention; it's the bedrock of mathematical consistency and clarity. In the expression at hand, we will scrupulously follow PEMDAS, meticulously working through each operation to reveal the final result. This exercise will not only yield the numerical answer but also reinforce our understanding of this crucial mathematical principle. From simple arithmetic problems to complex algebraic equations, the order of operations remains our unwavering compass, guiding us through the intricate world of numbers and symbols. So, let's begin, armed with PEMDAS, and conquer the challenge that lies before us.

Step-by-Step Evaluation

To evaluate this expression, we will meticulously follow the order of operations (PEMDAS) to ensure accuracy and clarity in our solution.

1. Parentheses

Our first step involves addressing the expression within the parentheses: $(-8 - 4)$. This is a straightforward subtraction of two negative numbers. When we subtract a number, it's the same as adding its negative. So, $-8 - 4$ is equivalent to $-8 + (-4)$. Adding two negative numbers results in a negative sum, and in this case, the sum is -12. Therefore, we have:

$(-8 - 4) = -12$

Now, let's replace the parentheses expression with its simplified value in the original expression. This substitution is a crucial step in maintaining the integrity of the equation while progressing towards the final answer. By simplifying within the parentheses first, we reduce the complexity of the overall expression, making subsequent calculations more manageable. This methodical approach aligns perfectly with the principles of PEMDAS, where parentheses take precedence. The ability to simplify expressions within parentheses is a fundamental skill in algebra and beyond, allowing us to break down complex problems into smaller, more easily digestible steps. This principle holds true whether we're dealing with simple arithmetic or intricate algebraic equations. So, with the parentheses neatly resolved, we can now proceed to the next stage of our evaluation, carrying forward the simplified value of -12 and continuing our journey towards the ultimate solution. The substitution is not just a mechanical step; it's a testament to the power of systematic problem-solving, a cornerstone of mathematical thinking.

2. Exponents

Next, we handle the exponent in the expression: $(-3)^2$. Remember that an exponent indicates repeated multiplication. In this case, $(-3)^2$ means $(-3)$ multiplied by itself. When we multiply two negative numbers, the result is a positive number. So, $(-3) * (-3)$ equals 9. Therefore:

$(-3)^2 = 9$

The concept of exponents is fundamental in mathematics, and understanding how to handle negative numbers within exponents is crucial. The rule that a negative number raised to an even power results in a positive number is a cornerstone of algebraic manipulation. This principle extends beyond simple arithmetic and plays a vital role in various mathematical fields, including calculus and complex numbers. Grasping the intricacies of exponents allows us to solve equations, model growth and decay, and explore the world of scientific notation. In our current expression, resolving the exponent is a pivotal step, transforming the term $(-3)^2$ into the manageable value of 9. This simplification paves the way for the next operations in the order of PEMDAS. The ability to accurately calculate exponents is not just a matter of memorizing rules; it's about understanding the underlying concept of repeated multiplication and its implications. With the exponent now addressed, we are one step closer to unraveling the complete solution, demonstrating the power of applying mathematical principles systematically.

3. Multiplication

Now we address the multiplication operation: $6(-3)^2$ which now translates to $6 * 9$. Multiplying 6 by 9 gives us 54. So:

$6 * 9 = 54$

Multiplication is a core arithmetic operation that forms the basis for many mathematical concepts. It's not just about repeated addition; it's a fundamental building block for understanding scaling, ratios, and proportions. In the context of our expression, the multiplication of 6 and 9 is a key step in simplifying the equation. This operation combines the result of the exponent evaluation with the coefficient, bringing us closer to a single numerical value. Mastering multiplication is essential not only for basic calculations but also for more advanced mathematical topics such as algebra, calculus, and statistics. The ability to quickly and accurately perform multiplication allows us to navigate complex equations with ease. In this particular instance, the product of 6 and 9, which is 54, is a significant milestone in our evaluation process. It represents a substantial portion of the overall expression, and its correct calculation is crucial for arriving at the final answer. With the multiplication step completed, we move on to the remaining operations, armed with this new piece of the puzzle. The journey through the order of operations continues, each step building upon the previous one, demonstrating the interconnectedness of mathematical processes.

4. Substitution

Let's substitute the values we've calculated back into the original expression. The expression now looks like this:

$54 - 24 - (-12)$

Substitution is a powerful technique in mathematics that allows us to replace parts of an expression with their simplified values. This process is crucial for breaking down complex problems into smaller, more manageable steps. In our case, we've replaced the original terms involving exponents and parentheses with their numerical equivalents, making the expression significantly easier to solve. Substitution is not just a mechanical act; it's a strategic tool that enhances clarity and reduces the risk of errors. By systematically substituting values, we maintain the integrity of the equation while progressing towards the final answer. This technique is widely used in algebra, calculus, and other advanced mathematical fields. The ability to substitute accurately and efficiently is a hallmark of strong mathematical reasoning. In our current problem, the substitution of 54 for $6(-3)^2$ and -12 for (-8 - 4) has transformed the expression into a simpler form involving only subtraction. This transformation sets the stage for the final steps in the evaluation process. The power of substitution lies in its ability to streamline complex expressions, making them more accessible and understandable. As we move forward, we will continue to rely on this technique to navigate the remaining operations and arrive at the ultimate solution.

5. Subtraction (and Addition)

Now, we perform the subtraction operations from left to right. First, we have $54 - 24$, which equals 30. So the expression becomes:

$30 - (-12)$

Next, we subtract a negative number, which is the same as adding its positive counterpart. Therefore, $30 - (-12)$ is equivalent to $30 + 12$, which equals 42. Hence:

$30 - (-12) = 42$

Subtraction and addition are fundamental arithmetic operations that are essential for problem-solving in mathematics and everyday life. Understanding how to handle subtraction, especially when dealing with negative numbers, is crucial for accurate calculations. Subtracting a negative number is a concept that often requires careful attention, as it is equivalent to adding the positive counterpart. This principle is not just a mathematical rule; it's a reflection of the way numbers interact on the number line. In our expression, the subtraction of -12 from 30 transforms into the addition of 12, leading us to the result of 42. This step highlights the importance of understanding the relationship between subtraction and addition, especially in the context of negative numbers. The ability to perform subtraction and addition fluently is a cornerstone of mathematical literacy. These operations are not only essential for basic arithmetic but also for more advanced topics such as algebra, calculus, and statistics. As we reach the final stages of our evaluation, the correct application of subtraction and addition principles brings us closer to the ultimate solution, demonstrating the power of mathematical reasoning.

Final Result

Therefore, the final value of the expression $6(-3)^2 - 24 - (-8 - 4)$ is 42.

This final result is the culmination of a series of meticulously executed steps, each guided by the principles of the order of operations. We began by simplifying the expression within the parentheses, then addressed the exponent, followed by multiplication, and finally, subtraction. Each step was crucial in transforming the original expression into its simplest numerical form. The final answer, 42, represents the single, definitive value of the expression. This process demonstrates the power of systematic problem-solving in mathematics. By breaking down a complex expression into smaller, more manageable steps, we can navigate intricate calculations with confidence and accuracy. The journey from the initial expression to the final result is a testament to the elegance and precision of mathematics. Each operation, each substitution, each simplification plays a vital role in the overall solution. The final answer is not just a number; it's the culmination of a logical and methodical approach, showcasing the beauty and consistency of mathematical principles. This exercise not only provides a numerical answer but also reinforces our understanding of the order of operations and its importance in mathematical problem-solving.

Conclusion

In conclusion, evaluating the expression $6(-3)^2 - 24 - (-8 - 4)$ required a careful application of the order of operations (PEMDAS). By systematically addressing parentheses, exponents, multiplication, and subtraction, we arrived at the final answer of 42. This exercise underscores the importance of precision and attention to detail in mathematical calculations. Mastering the order of operations is essential for success in mathematics and related fields.

The ability to navigate complex expressions and arrive at the correct solution is a hallmark of mathematical proficiency. It's not just about memorizing rules; it's about understanding the underlying principles and applying them strategically. The expression we evaluated today is a microcosm of the broader mathematical landscape, where the order of operations serves as a guiding star. From simple arithmetic to complex algebraic equations, this principle remains unwavering, ensuring consistency and accuracy. The journey through this particular expression has reinforced the importance of each step, from simplifying within parentheses to addressing exponents and performing multiplication and subtraction in the correct sequence. The final answer, 42, is more than just a number; it's a symbol of our ability to solve problems systematically and effectively. This exercise is a reminder that mathematics is not just about calculations; it's about logical reasoning, critical thinking, and the pursuit of precise solutions. As we move forward in our mathematical endeavors, the lessons learned from this evaluation will serve as a valuable foundation, empowering us to tackle more complex challenges with confidence and skill. The world of mathematics is vast and intricate, but with a solid understanding of fundamental principles, we can navigate its complexities and unlock its beauty.