Step-by-Step Solution: (5 + 2) × 3 - (2 + 5) × 3

Aug 3, 2025 by Viktoria Ivanova 49 views

Unraveling the Expression (5 + 2) × 3 - (2 + 5) × 3: A Step-by-Step Guide

Hey guys! Ever stumbled upon a mathematical expression that looks like a jumbled mess of numbers and symbols? Fear not! We're here to break down the expression (5 + 2) × 3 - (2 + 5) × 3 into digestible steps, making it super easy to understand. So, grab your thinking caps, and let's dive into this mathematical adventure!

Understanding the Order of Operations

Before we even think about crunching those numbers, it's crucial to understand the golden rule of mathematics: the order of operations. Think of it as the recipe for solving any mathematical expression. We follow the acronym PEMDAS/BODMAS, which stands for:

Parentheses / Brackets
Exponents / Orders
Multiplication and Division (from left to right)
Addition and Subtraction (from left to right)

This order dictates the sequence in which we perform operations. So, parentheses first, then exponents, and so on. Ignoring this order can lead to some seriously wrong answers, and we definitely don't want that!

When dealing with complex expressions, understanding the order of operations is paramount. This mathematical convention, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), ensures consistency and accuracy in calculations. By adhering to this order, we can systematically simplify expressions, breaking them down into manageable steps. Failing to respect the order of operations can lead to incorrect results, underscoring its importance in mathematical problem-solving. The principle behind PEMDAS is to prioritize operations that have a greater impact on the outcome of the expression, such as parentheses and exponents, before moving on to less impactful operations like addition and subtraction. This structured approach helps to eliminate ambiguity and guarantees that everyone arrives at the same solution, regardless of who is performing the calculation. So, remember, PEMDAS is your guide to navigating the world of mathematical expressions! Mastering the order of operations not only helps in solving mathematical problems but also enhances logical thinking and problem-solving skills in various aspects of life. By understanding the precedence of different operations, one can approach complex tasks in a structured and efficient manner, breaking them down into smaller, more manageable steps. This skill is invaluable not just in mathematics but also in computer programming, engineering, and many other fields where logical reasoning is essential. So, whether you're balancing your checkbook or designing a bridge, the principles of PEMDAS can help you achieve accurate and reliable results. Always keep this essential rule in mind: when faced with a complex expression, pause, recall the order of operations, and proceed with confidence.

Step 1: Tackling the Parentheses

Alright, let's get our hands dirty! Looking at our expression, (5 + 2) × 3 - (2 + 5) × 3, the first thing that screams at us are the parentheses. According to PEMDAS/BODMAS, we gotta deal with those first.

Inside the first set of parentheses, we have (5 + 2), which is a simple addition. 5 plus 2 equals 7. Easy peasy!

Now, let's move on to the second set of parentheses, (2 + 5). Guess what? It's the same numbers, just in a different order! So, 2 plus 5 also equals 7. Who would've thought?

Our expression now looks like this: 7 × 3 - 7 × 3. We've successfully conquered the parentheses, and we're one step closer to solving this puzzle.

Parentheses, denoted by the symbols ( ) in mathematical expressions, serve as vital tools for grouping operations and dictating the order in which they are performed. In the expression (5 + 2) × 3 - (2 + 5) × 3, the parentheses clearly delineate the addition operations that must be carried out before any other operations are considered. This use of parentheses ensures that the expression is evaluated in a precise and unambiguous manner. Without parentheses, the order of operations might lead to a different interpretation and an incorrect result. For instance, if we were to omit the parentheses and simply write 5 + 2 × 3, the multiplication would take precedence over the addition, potentially leading to a different outcome. By enclosing the addition operations within parentheses, we explicitly instruct the calculator (or the person solving the equation) to perform these operations first. This not only simplifies the expression but also reduces the risk of error. Furthermore, parentheses can be nested within each other, creating layers of operations that must be resolved sequentially, starting from the innermost set. This hierarchical structure allows for the creation of complex expressions that can be systematically simplified by carefully following the order of operations. In addition to their role in specifying the order of operations, parentheses also enhance the readability of expressions. By visually grouping terms and operations, parentheses make it easier to understand the structure of the expression and the relationships between its components. This clarity is particularly important in complex equations with multiple operations and variables. So, always remember to pay close attention to parentheses when solving mathematical expressions, as they hold the key to unlocking the correct answer.

Step 2: Multiplication Time!

Next up on our PEMDAS/BODMAS adventure is multiplication. We've got two multiplication operations staring us right in the face: 7 × 3 and 7 × 3. Let's tackle them one by one.

7 multiplied by 3 is 21. Simple enough, right?

And guess what? 7 multiplied by 3 again is, you guessed it, 21!

Our expression has now transformed into: 21 - 21. We're getting closer to the finish line, guys!

Multiplication is a fundamental arithmetic operation that represents the repeated addition of a number. In the expression (5 + 2) × 3 - (2 + 5) × 3, multiplication plays a crucial role in scaling the results of the addition operations within the parentheses. After simplifying the parentheses to obtain 7 × 3 - 7 × 3, the multiplication operations become the next logical step according to the order of operations. Multiplying 7 by 3 yields 21, effectively tripling the value obtained from the addition. This multiplication step is essential for determining the overall value of the expression and cannot be skipped or performed out of order. Understanding the concept of multiplication as repeated addition can help visualize its effect on the numbers involved. For instance, 7 × 3 can be thought of as adding 7 to itself three times (7 + 7 + 7), resulting in 21. This perspective can be particularly helpful in grasping the magnitude of the result and ensuring its accuracy. In more complex expressions, multiplication may involve multiple terms and variables, each contributing to the final outcome. Therefore, mastering the rules and techniques of multiplication is essential for success in mathematics and related fields. Additionally, multiplication has various applications in real-world scenarios, such as calculating areas, volumes, and proportions. Whether you're determining the amount of paint needed to cover a wall or calculating the cost of a bulk purchase, multiplication is an indispensable tool. So, embrace the power of multiplication and let it guide you through the intricate world of numbers!

Step 3: The Grand Finale - Subtraction

Drumroll, please! We've reached the final step: subtraction. Our expression is now a simple 21 - 21.

What's 21 minus 21? Zero! That's right, the answer is 0.

We did it! We successfully unraveled the expression (5 + 2) × 3 - (2 + 5) × 3, and the result is 0. Give yourselves a pat on the back, you mathematical wizards!

Subtraction, the inverse operation of addition, is the process of finding the difference between two numbers. In the expression (5 + 2) × 3 - (2 + 5) × 3, subtraction serves as the final operation, combining the results of the preceding steps to arrive at the solution. After performing the parentheses and multiplication operations, we are left with 21 - 21, a straightforward subtraction problem. Subtracting 21 from 21 yields 0, indicating that the two quantities are equal and cancel each other out. This result highlights the importance of understanding the properties of mathematical operations, such as the additive inverse, which states that for every number a, there exists a number -a such that a + (-a) = 0. In this case, 21 and -21 are additive inverses, and their sum is 0. Subtraction is a fundamental operation in mathematics with applications in various fields, including finance, physics, and engineering. From calculating the change in a bank account to determining the distance between two points, subtraction is an indispensable tool for solving real-world problems. Mastering the techniques of subtraction is essential for developing a strong foundation in mathematics and for confidently tackling more complex calculations. Additionally, subtraction can be visualized using number lines or other graphical representations, which can aid in understanding the concept and its applications. So, embrace the power of subtraction and let it be your guide in navigating the world of numbers and quantities!

Conclusion: Math Can Be Fun!

So, there you have it! We've taken a seemingly complex expression and broken it down into simple, manageable steps. Remember, the key to success in mathematics is understanding the order of operations and tackling problems one step at a time. And hey, if we can do it, you can too! Keep practicing, keep exploring, and most importantly, keep having fun with math! You got this!