Solving (3 × 2²) ÷ (6 - 4): A Step-by-Step Guide

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Let's break down how to solve the mathematical expression (3 × 2²) ÷ (6 - 4) in a clear and easy-to-understand way. Mathematics can sometimes seem daunting, but by following the order of operations, we can tackle even complex problems with confidence. This guide will walk you through each step, ensuring you grasp the fundamental principles involved. So, let’s dive in and unravel this equation together!

Understanding the Order of Operations (PEMDAS/BODMAS)

Before we begin, it’s crucial to understand the order of operations, often remembered by the acronyms PEMDAS or BODMAS. This order dictates the sequence in which mathematical operations should be performed to arrive at the correct answer.

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

Knowing and applying PEMDAS/BODMAS is the cornerstone of solving mathematical expressions accurately. It ensures consistency and avoids ambiguity in calculations. Without a standardized order, different people might interpret the same expression in various ways, leading to different results. Think of it as a universal language for math, ensuring everyone is on the same page. Mastering this order will not only help you solve this specific problem but also equip you with the skills to tackle more complex mathematical challenges in the future. So, let's keep this order in mind as we proceed to solve the given expression.

Step 1: Simplify within the Parentheses

Our first step, guided by PEMDAS/BODMAS, is to simplify the expression within the parentheses: (6 - 4). This is a straightforward subtraction, and it's essential to tackle it first. Inside the parentheses, we have 6 minus 4. Performing this subtraction gives us a result of 2. So, (6 - 4) simplifies to 2. Now, our expression looks like this: (3 × 2²) ÷ 2.

Simplifying within parentheses is crucial because it isolates and resolves a portion of the equation before it interacts with the rest. This approach reduces complexity and prevents errors that could arise from prematurely mixing different operations. Think of it as solving a mini-puzzle within the larger mathematical puzzle. By addressing the parentheses first, we create a simpler, more manageable equation to work with. This step highlights the importance of meticulousness in mathematics – taking each component one at a time to ensure accuracy and clarity. So, with the parentheses simplified, we're ready to move on to the next step in our calculation, which involves dealing with exponents.

Step 2: Evaluate the Exponent

Following the order of operations, the next step is to evaluate the exponent. In our expression, (3 × 2²) ÷ 2, we have 2², which means 2 raised to the power of 2. This is the same as multiplying 2 by itself: 2 × 2. Performing this calculation, we find that 2² equals 4. Now our expression becomes (3 × 4) ÷ 2.

Exponents represent repeated multiplication, and evaluating them is a critical step in simplifying mathematical expressions. They indicate how many times a base number is multiplied by itself. Understanding and correctly calculating exponents is fundamental to numerous mathematical concepts, including algebra, geometry, and calculus. In this specific problem, evaluating 2² to 4 simplifies the expression and allows us to proceed with the remaining operations. Failing to address the exponent at this stage would lead to an incorrect result, emphasizing the importance of adhering to the order of operations. So, with the exponent evaluated, we're now ready to tackle the multiplication and division operations in our expression.

Step 3: Perform Multiplication

Now that we've handled the parentheses and the exponent, it's time to move on to multiplication. In our simplified expression, (3 × 4) ÷ 2, we have a multiplication operation: 3 × 4. Multiplying 3 by 4 gives us 12. So, our expression now looks like this: 12 ÷ 2.

Multiplication is a fundamental arithmetic operation that represents the repeated addition of a number. In the context of the order of operations, multiplication is performed before addition and subtraction but after parentheses and exponents. Accurately performing the multiplication step is crucial for arriving at the correct final answer. In this case, 3 multiplied by 4 equals 12, which simplifies our expression further and brings us closer to the solution. It's important to remember that multiplication and division have equal priority in the order of operations, so they are performed from left to right. With the multiplication completed, we are now left with a simple division problem to solve.

Step 4: Perform Division

Our final step is to perform the division. We now have the expression 12 ÷ 2. Dividing 12 by 2 is a straightforward operation. When we divide 12 by 2, we get 6. Thus, the final result of our expression is 6.

Division, the inverse operation of multiplication, is the process of splitting a number into equal groups. In the order of operations, division holds the same priority as multiplication and is performed from left to right. In this particular problem, dividing 12 by 2 gives us the final answer of 6. This concludes the simplification of our expression, and we have successfully navigated through each step, adhering to the order of operations. This final step underscores the importance of understanding basic arithmetic operations and their roles in solving mathematical expressions. With the division completed, we have arrived at the solution, demonstrating the power of methodical problem-solving in mathematics.

Final Answer

Therefore, the solution to the mathematical expression (3 × 2²) ÷ (6 - 4) is 6. By following the order of operations (PEMDAS/BODMAS) meticulously, we were able to break down the problem into manageable steps and arrive at the correct answer. Remember, understanding and applying these principles is key to success in mathematics. This step-by-step approach not only helps in solving this specific problem but also builds a strong foundation for tackling more complex mathematical challenges in the future. Keep practicing and reinforcing these concepts to enhance your mathematical skills and confidence.

In conclusion, mastering the order of operations and practicing regularly are essential for solving mathematical expressions accurately. By breaking down complex problems into smaller, manageable steps, we can approach any equation with confidence. If you're interested in further exploring mathematical concepts and practice problems, Khan Academy is an excellent resource.