Solving $[(3+5) × 2-5] × 3$: A Step-by-Step Guide

by Alex Johnson 50 views

Let's dive into the world of mathematics and break down this intriguing expression: [(3+5)×2−5]×3[(3+5) × 2-5] × 3. In this comprehensive guide, we'll walk through each step, ensuring you not only understand the solution but also grasp the underlying principles. Whether you're a student looking to ace your math exams or simply a curious mind eager to sharpen your skills, this article is tailored just for you.

Understanding the Order of Operations

At the heart of solving any mathematical expression lies the order of operations, often remembered by the acronym PEMDAS or BODMAS. This mnemonic helps us prioritize which operations to perform first. PEMDAS stands for:

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

BODMAS, commonly used in some regions, stands for:

  • Brackets
  • Orders
  • Division and Multiplication (from left to right)
  • Addition and Subtraction (from left to right)

Both acronyms essentially convey the same hierarchy, ensuring consistent results when evaluating mathematical expressions. By adhering to this order, we can systematically break down complex problems into manageable steps, minimizing errors and maximizing clarity. This foundation is crucial for anyone looking to build confidence in their mathematical abilities.

Step 1: Tackling the Parentheses

Our mathematical journey begins with the expression inside the parentheses: (3+5)(3+5). According to the order of operations, anything within parentheses or brackets takes precedence. This is because parentheses group terms together, indicating that these operations should be performed as a single unit before anything else. Ignoring this rule can lead to incorrect results, emphasizing the importance of following the correct sequence.

Inside our parentheses, we have a simple addition problem. Adding 3 and 5 is a fundamental arithmetic operation that most people are familiar with. It forms the building block for more complex calculations. By starting with this straightforward step, we set the stage for the rest of the problem, making the overall solution process smoother and more logical. This initial step not only simplifies the expression but also reinforces the importance of basic arithmetic skills in more advanced mathematical contexts.

So, let's perform the addition:

3+5=83 + 5 = 8

Now, our expression looks like this:

[8imes2−5]imes3[8 imes 2 - 5] imes 3

By completing the operation within the parentheses, we've taken the first step towards simplifying the entire expression. This methodical approach, breaking down the problem into smaller, manageable chunks, is a key strategy for success in mathematics. Each step builds upon the previous one, creating a clear path to the final answer. This approach is particularly useful when dealing with longer or more complex equations, where the risk of error increases without a systematic method.

Step 2: Multiplication Within the Brackets

Now that we've simplified the parentheses, our attention shifts to the brackets: [8imes2−5][8 imes 2 - 5]. Within these brackets, we encounter two operations: multiplication and subtraction. According to the order of operations, multiplication takes precedence over subtraction. This means we must perform the multiplication before we can subtract. Prioritizing multiplication ensures that we adhere to the established rules of mathematics and maintain the integrity of the equation.

Multiplying 8 by 2 is another fundamental arithmetic operation. Mastering these basic operations is crucial for tackling more advanced mathematical concepts. Multiplication is the process of adding a number to itself a certain number of times, and in this case, we are adding 8 to itself twice. The result of this multiplication will be used in the next step of our calculation, highlighting the sequential nature of mathematical problem-solving.

So, let's multiply:

8imes2=168 imes 2 = 16

Our expression now transforms into:

[16−5]imes3[16 - 5] imes 3

By performing the multiplication, we've further simplified the expression within the brackets. This step-by-step simplification is a hallmark of effective mathematical problem-solving. By focusing on one operation at a time, we reduce the complexity of the problem and minimize the chances of making mistakes. This methodical approach also allows for easier checking of our work, ensuring accuracy and confidence in our solution.

Step 3: Subtraction Inside the Brackets

With the multiplication completed, we now focus on the remaining operation within the brackets: [16−5][16 - 5]. We have a simple subtraction problem to tackle. Subtraction is one of the basic arithmetic operations, representing the process of taking away one number from another. It is a fundamental concept that is used extensively in various mathematical and real-world scenarios.

Subtracting 5 from 16 is a straightforward calculation. It helps us reduce the expression within the brackets to a single value. This step is crucial because it prepares us for the final operation outside the brackets. By isolating the result of this subtraction, we simplify the overall equation, making it easier to manage and solve. This methodical reduction of complexity is a key skill in mathematics, allowing us to approach problems with clarity and precision.

Let's perform the subtraction:

16−5=1116 - 5 = 11

Our expression now looks like this:

11imes311 imes 3

By completing the subtraction, we have successfully simplified the expression inside the brackets to a single number. This step highlights the importance of working systematically through the order of operations, reducing complexity at each stage. This methodical approach not only makes the problem easier to solve but also provides a clear and logical pathway to the final answer. It reinforces the idea that complex mathematical problems can be broken down into simpler, more manageable steps.

Step 4: Final Multiplication

We've reached the final step of our mathematical journey! Our expression has been simplified to: 11imes311 imes 3. Now, we just need to perform this final multiplication to arrive at the solution. This is the last operation in our problem, and it brings together all the previous steps to give us our final result. Multiplication, as we discussed earlier, is the process of adding a number to itself a certain number of times. In this case, we are multiplying 11 by 3, which means adding 11 to itself three times.

Multiplying 11 by 3 is a basic arithmetic operation, but it's crucial for completing our calculation. The result of this multiplication will be the final answer to the problem we set out to solve. This step underscores the importance of mastering fundamental arithmetic skills, as they form the basis for more complex mathematical operations. By performing this final multiplication accurately, we validate all the steps we've taken so far and demonstrate our understanding of the order of operations.

Let's multiply:

11imes3=3311 imes 3 = 33

Therefore, the final answer to our expression [(3+5)×2−5]×3[(3+5) × 2-5] × 3 is 33.

Conclusion

In this step-by-step guide, we've successfully navigated the mathematical expression [(3+5)×2−5]×3[(3+5) × 2-5] × 3. By meticulously following the order of operations and breaking down the problem into smaller, manageable steps, we arrived at the solution: 33. Remember, the key to mastering mathematics is understanding the fundamental principles and applying them systematically. Keep practicing, and you'll find that even the most complex problems can be solved with confidence. For more resources on mathematical problem-solving, visit reputable websites like Khan Academy.