Solving Mathematical Expressions A Step-by-Step Guide To (15 × 3 - 3) × (9 + 6)
In the realm of mathematics, expressions often appear as puzzles waiting to be solved. These mathematical puzzles, composed of numbers, operators, and parentheses, hold a certain allure for those who enjoy deciphering their hidden values. Today, we embark on a journey to unravel the intricacies of the expression (15 × 3 - 3) × (9 + 6). Through a step-by-step exploration, we will not only arrive at the final answer but also gain a deeper appreciation for the fundamental principles that govern mathematical operations.
The Order of Operations: A Guiding Light
Before we dive into the specifics of our expression, it is crucial to understand the order of operations, a set of rules that dictate the sequence in which mathematical operations should be performed. This order, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), ensures that mathematical expressions are evaluated consistently, regardless of who is solving them. By adhering to this order, we can avoid ambiguity and arrive at the correct result.
In the context of our expression, (15 × 3 - 3) × (9 + 6), the order of operations provides a clear roadmap for our calculations. We must first address the operations within the parentheses, followed by any multiplication or division, and finally, addition or subtraction. This structured approach guarantees that we accurately navigate the expression and arrive at the intended solution. Let’s break down each step with detailed explanations to ensure clarity and understanding.
Step 1: Tackling the Parentheses
Our expression, (15 × 3 - 3) × (9 + 6), features two sets of parentheses, each containing its own set of operations. According to the order of operations, we must first simplify the expressions within these parentheses before proceeding with any other calculations. This initial step is crucial as it sets the stage for the rest of our problem-solving process.
Inside the First Parentheses: (15 × 3 - 3)
The first set of parentheses, (15 × 3 - 3), presents us with a combination of multiplication and subtraction. Following the order of operations, we must perform the multiplication before the subtraction. This means we first multiply 15 by 3, which equals 45. Then, we subtract 3 from the result, giving us 42. Thus, the simplified value of the first set of parentheses is 42. This step exemplifies the importance of prioritizing operations to maintain accuracy in mathematical computations.
Inside the Second Parentheses: (9 + 6)
The second set of parentheses, (9 + 6), is more straightforward, involving only addition. Adding 9 and 6 together, we arrive at a sum of 15. Therefore, the simplified value of the second set of parentheses is 15. This simple addition highlights how the order of operations ensures that even basic calculations are performed in the correct sequence, contributing to the overall precision of the mathematical solution.
With both sets of parentheses now simplified, our expression transforms from (15 × 3 - 3) × (9 + 6) to 42 × 15. This transformation marks a significant milestone in our problem-solving journey, setting the stage for the final calculation that will reveal the ultimate answer. By methodically addressing the parentheses first, we have effectively streamlined the expression, making it easier to manage and solve.
Step 2: The Final Multiplication
With the parentheses now resolved, our expression has been simplified to a single multiplication: 42 × 15. This step represents the culmination of our step-by-step approach, where we bring together the results from the previous calculations to arrive at the final answer. Multiplying 42 by 15 is a straightforward process, but it is essential to carry it out carefully to ensure accuracy.
To perform the multiplication, we can use the standard multiplication algorithm. Multiplying 42 by 5 (the units digit of 15) gives us 210. Then, multiplying 42 by 10 (the tens digit of 15) gives us 420. Adding these two results together, 210 + 420, yields a sum of 630. Thus, the product of 42 and 15 is 630. This final calculation demonstrates how the individual steps, each guided by the order of operations, converge to provide a complete and correct solution.
Therefore, the final answer to the expression (15 × 3 - 3) × (9 + 6) is 630. This result encapsulates the entire process we have undertaken, from breaking down the initial expression to methodically simplifying each component. The journey to this solution underscores the power of structured problem-solving in mathematics, where each step builds upon the previous one to achieve a cohesive and accurate outcome.
The Significance of Mathematical Expressions
Mathematical expressions, such as the one we have just solved, are more than just collections of numbers and symbols. They are the language of mathematics, allowing us to represent and solve a wide range of problems in various fields. From simple arithmetic to complex engineering calculations, mathematical expressions form the foundation of quantitative reasoning and problem-solving.
Understanding how to manipulate and solve these expressions is a fundamental skill in mathematics education. It equips students with the ability to think critically, analyze problems, and develop logical solutions. The process of simplifying an expression, such as (15 × 3 - 3) × (9 + 6), involves applying mathematical principles and following a systematic approach. This not only leads to the correct answer but also reinforces the importance of precision and attention to detail.
Moreover, mathematical expressions serve as building blocks for more advanced mathematical concepts. As students progress in their mathematical journey, they will encounter more complex expressions that require a solid understanding of the order of operations and other fundamental principles. The ability to confidently navigate these expressions is crucial for success in higher-level mathematics and related fields.
Real-World Applications of Mathematical Expressions
The relevance of mathematical expressions extends far beyond the classroom. In the real world, these expressions are used in countless applications, ranging from everyday calculations to sophisticated scientific models. Whether it is calculating the cost of groceries, determining the trajectory of a rocket, or analyzing financial data, mathematical expressions play a vital role.
In fields such as engineering, physics, and computer science, mathematical expressions are indispensable tools for modeling and simulating real-world phenomena. Engineers use expressions to design structures and machines, physicists use them to describe the laws of nature, and computer scientists use them to create algorithms and software. The ability to formulate and solve mathematical expressions is therefore a highly valued skill in these professions.
Even in seemingly non-mathematical fields, expressions can be found lurking beneath the surface. For example, in economics, supply and demand curves are often represented using mathematical equations, which can be used to predict market behavior. In healthcare, mathematical models are used to study the spread of diseases and to optimize treatment strategies. The pervasive nature of mathematical expressions highlights their importance in our modern world.
Conclusion: Mastering Mathematical Expressions
In conclusion, the expression (15 × 3 - 3) × (9 + 6) provides a valuable example of how mathematical principles and the order of operations work together to produce a solution. By breaking down the expression into smaller, manageable steps, we were able to systematically simplify it and arrive at the final answer of 630. This process underscores the importance of following a structured approach when solving mathematical problems.
Furthermore, the significance of mathematical expressions extends beyond the realm of pure mathematics. These expressions are essential tools for problem-solving in a wide range of fields, from science and engineering to finance and economics. Mastering the art of manipulating and solving mathematical expressions is therefore a valuable skill that can open doors to numerous opportunities.
As we continue our exploration of mathematics, it is crucial to remember that every expression, no matter how complex it may seem, can be tackled with a combination of knowledge, patience, and a systematic approach. By embracing the challenges presented by mathematical expressions, we not only enhance our problem-solving abilities but also gain a deeper appreciation for the beauty and power of mathematics.
