Solving $51-\left\{\left(3^2-2\right)+(-3+\sqrt{49})-4\right\} * 10$ A Step-by-Step Guide

Introduction

In the realm of mathematics, problem-solving stands as a cornerstone of understanding and application. This article delves into the intricate process of solving the mathematical expression $51-\left\{\left(3^2-2\right)+(-3+\sqrt{49})-4\right\} * 10$, providing a step-by-step guide to unraveling its solution. Our mathematical journey will not only elucidate the correct answer but also emphasize the importance of adhering to the order of operations and the fundamental principles governing arithmetic expressions. We aim to transform this complex equation into a comprehensible narrative, making it accessible to both math enthusiasts and those seeking to enhance their problem-solving acumen. By breaking down each step, we hope to demystify the process and highlight the elegance inherent in mathematical solutions. This in-depth exploration will serve as a testament to the power of methodical thinking and the beauty of mathematical precision. As we navigate through this equation, we'll encounter various mathematical concepts, from exponents and square roots to addition, subtraction, and multiplication, each playing a crucial role in arriving at the final answer. So, let's embark on this mathematical adventure and unlock the solution to this intriguing problem.

Dissecting the Expression: A Step-by-Step Approach

The cornerstone of solving any mathematical expression lies in understanding and applying the correct order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). In our quest to solve $51-\left\{\left(3^2-2\right)+(-3+\sqrt{49})-4\right\} * 10$, we will meticulously follow this order, dissecting the expression step by step. First, we'll tackle the parentheses, which contain a mix of exponents, square roots, and basic arithmetic operations. Within the innermost parentheses, $3^2$ takes precedence, resolving to 9. Then, we subtract 2, yielding 7. Next, we address the square root of 49, which equals 7. Adding this to -3 gives us 4. Now, we can simplify the expression inside the curly braces: $(7 + 4 - 4)$. This simplifies to 7, setting the stage for the next operation. Following the parentheses, we move on to multiplication. We multiply the simplified expression within the curly braces (7) by 10, resulting in 70. Finally, we perform the subtraction: $51 - 70$. This step-by-step approach not only ensures accuracy but also provides a clear roadmap for understanding the solution. By methodically breaking down the expression, we eliminate ambiguity and pave the way for a precise answer. This systematic approach is the key to mastering complex mathematical problems.

Navigating Parentheses and Braces: The Core of the Equation

To accurately solve the mathematical expression, a crucial aspect involves effectively navigating parentheses and braces. These symbols dictate the order in which operations are performed, and a misstep can lead to an incorrect solution. Our expression, $51-\left\{\left(3^2-2\right)+(-3+\sqrt{49})-4\right\} * 10$, presents a nested structure of parentheses and braces, demanding a careful approach. The innermost parentheses, $\left(3^2-2\right)$ and $(-3+\sqrt{49})$, are our starting point. Within the first set, we address the exponent, $3^2$, which equals 9. Subtracting 2 from 9 gives us 7. In the second set, we first evaluate the square root of 49, which is 7. Adding this to -3 yields 4. Now, we substitute these results back into the expression, simplifying it to $51-\{7+4-4\} * 10$. The curly braces now enclose the simplified expression $7 + 4 - 4$. Performing the addition and subtraction from left to right, we get $11 - 4$, which equals 7. The expression is further simplified to $51 - 7 * 10$. This meticulous handling of parentheses and braces ensures that we adhere to the correct order of operations, setting the stage for the subsequent steps. The accurate navigation of these symbols is not just a procedural requirement; it's a fundamental skill in mathematical problem-solving.

Exponents and Square Roots: Unveiling the Hidden Values

In the intricate dance of mathematical operations, exponents and square roots play pivotal roles, often unveiling hidden values within an expression. Our problem, $51-\left\{\left(3^2-2\right)+(-3+\sqrt{49})-4\right\} * 10$, features both, and their correct evaluation is crucial for arriving at the accurate solution. The term $3^2$ represents 3 raised to the power of 2, which means 3 multiplied by itself. This calculation yields 9, a key value in simplifying the expression within the parentheses. The square root, denoted by the symbol √, asks us to find a number that, when multiplied by itself, equals the number under the root. In our case, we have $\sqrt{49}$, which translates to finding the number that, when squared, gives 49. This number is 7, as $7 * 7 = 49$. By accurately evaluating the exponent and the square root, we transform the expression from a complex string of symbols into a more manageable form. These operations are not merely mechanical steps; they are about understanding the underlying mathematical concepts. The correct interpretation and calculation of exponents and square roots are essential skills in mathematics, enabling us to unlock the hidden values and progress towards the solution.

Multiplication and Subtraction: The Final Act

With the parentheses, exponents, and square roots resolved, we arrive at the final act of our mathematical play: multiplication and subtraction. Our expression has been meticulously simplified to $51 - 7 * 10$. According to the order of operations (PEMDAS), multiplication takes precedence over subtraction. Thus, we first perform the multiplication: $7 * 10$, which equals 70. Now, the expression is reduced to a simple subtraction: $51 - 70$. Subtracting 70 from 51 results in -19. This final step showcases the elegance of mathematical precision, where each operation leads us closer to the definitive answer. The order in which we perform these operations is paramount; a deviation from the established rules would lead to a different, and incorrect, result. The correct application of multiplication and subtraction not only provides the solution but also reinforces the importance of following mathematical conventions. This final act is a testament to the power of methodical calculation and the satisfaction of arriving at the correct answer.

The Solution: $51-\left\{\left(3^2-2\right)+(-3+\sqrt{49})-4\right\} * 10 = -19$

After a meticulous step-by-step journey through the complexities of the expression $51-\left\{\left(3^2-2\right)+(-3+\sqrt{49})-4\right\} * 10$, we arrive at the solution: -19. Our exploration began with a dissection of the expression, emphasizing the paramount importance of adhering to the order of operations (PEMDAS). We navigated the nested parentheses and braces, carefully evaluating exponents and square roots. Each step was a deliberate move, ensuring accuracy and clarity in our process. The accurate evaluation of $3^2$ as 9 and $\sqrt{49}$ as 7 were pivotal moments in simplifying the expression. We then executed the multiplication and subtraction in their correct order, culminating in the final answer. This journey underscores the significance of methodical problem-solving in mathematics. It's not just about arriving at the answer; it's about understanding the process, the logic, and the underlying principles. The solution, -19, is not merely a number; it's the culmination of a careful and deliberate mathematical exploration. This final result stands as a testament to the power of precise calculation and the beauty of mathematical reasoning.

Conclusion: Mastering the Art of Mathematical Problem-Solving

In conclusion, the journey to solve the mathematical expression $51-\left\{\left(3^2-2\right)+(-3+\sqrt{49})-4\right\} * 10$ has been a testament to the art of mathematical problem-solving. Through a systematic application of the order of operations, we successfully navigated the complexities of parentheses, exponents, square roots, multiplication, and subtraction. The final solution, -19, is not just a numerical answer; it's a symbol of the power of methodical thinking and the importance of precision in mathematics. This exploration has highlighted the significance of understanding fundamental mathematical principles and applying them diligently. Each step, from evaluating exponents to performing the final subtraction, required careful consideration and adherence to established rules. The ability to break down a complex problem into smaller, manageable steps is a crucial skill in mathematics, and this exercise has served as a practical demonstration of its effectiveness. Mastering the art of mathematical problem-solving is not just about finding the right answer; it's about developing a logical and analytical mindset that can be applied to a wide range of challenges. This comprehensive approach is what truly empowers us to unravel the mysteries of mathematics and beyond.