Solving 10 * 3^2 + (10 - 4) / 2 - √64 A Step-by-Step Guide

In the realm of mathematics, precision and order are paramount. Evaluating complex expressions requires a systematic approach, adhering to the established order of operations. This article delves into the step-by-step solution of the expression 10 ⋅ 3² + (10 - 4) / 2 - √64, providing a comprehensive understanding of the mathematical principles involved. We will break down each component, meticulously applying the rules of precedence to arrive at the correct answer. This exploration serves not only as a practical exercise but also as a reinforcement of fundamental mathematical concepts. Understanding the order of operations is crucial for success in various mathematical disciplines, from basic arithmetic to advanced calculus. The ability to accurately solve equations and expressions is a cornerstone of problem-solving skills, essential in both academic and real-world contexts. Let's embark on this journey of mathematical discovery, unraveling the intricacies of this expression and solidifying our grasp of the order of operations.

Deciphering the Expression: A Step-by-Step Guide

To accurately evaluate the expression 10 ⋅ 3² + (10 - 4) / 2 - √64, we must diligently follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This mnemonic device serves as a roadmap, guiding us through the sequence of calculations. First, we address any expressions within parentheses, then exponents, followed by multiplication and division (from left to right), and finally, addition and subtraction (also from left to right). Ignoring this order can lead to incorrect results, highlighting the importance of a structured approach. Each step in the process builds upon the previous one, gradually simplifying the expression until we arrive at the final answer. This methodical approach not only ensures accuracy but also fosters a deeper understanding of the underlying mathematical principles. Let's begin our step-by-step journey, carefully dissecting the expression and applying the rules of precedence.

1. Parentheses First: Simplifying (10 - 4)

According to PEMDAS, our initial focus should be on the expression enclosed within parentheses: (10 - 4). This is the first step in unraveling the complexity of the overall equation. Performing this subtraction is straightforward: 10 minus 4 equals 6. Thus, we replace the parenthetical expression with its simplified value, resulting in a new expression: 10 ⋅ 3² + 6 / 2 - √64. By addressing the parentheses first, we have effectively reduced the number of operations and simplified the equation, bringing us closer to the final solution. This step emphasizes the significance of prioritizing parentheses in the order of operations, as it often lays the groundwork for subsequent calculations. The result of this initial simplification serves as the foundation for the next step in our mathematical journey.

2. Tackling the Exponent: Evaluating 3²

With the parentheses addressed, the next priority in the order of operations is exponents. In our expression, we encounter the term 3², which signifies 3 raised to the power of 2. This is equivalent to 3 multiplied by itself: 3 * 3 = 9. Replacing 3² with its value, 9, transforms the expression into 10 ⋅ 9 + 6 / 2 - √64. By evaluating the exponent, we continue to streamline the equation, making it more manageable for the subsequent operations. This step underscores the importance of exponents in mathematical expressions and their position within the hierarchy of operations. Accurately calculating exponents is crucial for obtaining the correct final result. The simplified expression now sets the stage for addressing multiplication and division.

3. Multiplication and Division: 10 ⋅ 9 and 6 / 2

Following the order of operations, we now turn our attention to multiplication and division. These operations hold equal precedence and are performed from left to right. Our expression presents us with two such operations: 10 ⋅ 9 and 6 / 2. First, we tackle the multiplication: 10 multiplied by 9 equals 90. Next, we perform the division: 6 divided by 2 equals 3. Replacing these operations with their respective results, the expression now becomes 90 + 3 - √64. This step demonstrates the importance of addressing multiplication and division in the correct order, as altering the sequence can lead to discrepancies in the final answer. By systematically working through these operations, we are steadily progressing towards the solution.

4. The Square Root: Finding √64

Before we can proceed with addition and subtraction, we must address the square root. The term √64 represents the square root of 64, which is the number that, when multiplied by itself, equals 64. In this case, 8 multiplied by 8 equals 64, so √64 equals 8. Substituting 8 for √64 in our expression, we now have 90 + 3 - 8. This simplification brings us closer to the final calculation, as we are left with only addition and subtraction operations. Understanding square roots is essential in mathematics, and their proper evaluation is crucial for accurate results. By calculating the square root, we have further reduced the complexity of the expression and prepared it for the final steps.

5. Addition and Subtraction: 90 + 3 - 8

The final stage in solving our expression involves addition and subtraction. Similar to multiplication and division, these operations share equal precedence and are performed from left to right. We begin with the addition: 90 plus 3 equals 93. This gives us a new intermediate result: 93 - 8. Now, we perform the subtraction: 93 minus 8 equals 85. Therefore, the final result of the expression 10 ⋅ 3² + (10 - 4) / 2 - √64 is 85. This step highlights the importance of following the correct order, even in the final stages of calculation. By systematically applying addition and subtraction, we arrive at the definitive answer, completing our mathematical journey.

Conclusion: Mastering the Order of Operations

In conclusion, the evaluation of the expression 10 ⋅ 3² + (10 - 4) / 2 - √64 demonstrates the crucial role of the order of operations in mathematics. By meticulously following the PEMDAS rule, we successfully navigated the complexities of the expression, arriving at the final answer of 85. This exercise not only provides a concrete solution but also reinforces the fundamental principles of mathematical precedence. Mastering the order of operations is essential for accurate calculations in various mathematical contexts, from basic arithmetic to advanced problem-solving. The ability to confidently apply these rules ensures consistency and precision in mathematical endeavors. Through careful step-by-step analysis, we have unveiled the solution and strengthened our understanding of this vital mathematical concept.