Evaluate The Expression 4^2 + (2 X 5) X 3 - 6 ÷ 2

Evaluating mathematical expressions accurately requires a solid understanding of the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This principle ensures that everyone arrives at the same correct answer by performing operations in a consistent sequence. Let's delve into how we can break down and solve the expression $4^2 + (2 \times 5) \times 3 - 6 \div 2$, highlighting each step and the underlying rationale. The process not only reveals the final answer but also reinforces the importance of following mathematical conventions. By meticulously working through this expression, we’ll showcase the elegance and precision inherent in mathematical problem-solving. Grasping the order of operations is crucial not just for academic exercises, but also for various real-world applications where precise calculations are necessary. Whether it's calculating finances, engineering designs, or scientific analyses, understanding and applying PEMDAS is fundamental to achieving accurate results. Let’s begin this mathematical journey, dissecting each component of the expression to ensure clarity and precision in our evaluation. Remember, the beauty of mathematics lies in its logical structure and the ability to arrive at definitive solutions through systematic methods. This evaluation will demonstrate the step-by-step process, ensuring that every operation is performed in the correct order, ultimately leading us to the accurate answer. The significance of this exercise extends beyond the mere calculation; it's about honing our analytical skills and reinforcing the foundational principles of arithmetic. Mathematics is more than just numbers and symbols; it's a language of logic and precision that enables us to understand and model the world around us.

Step-by-Step Breakdown

Let's meticulously evaluate the expression $4^2 + (2 \times 5) \times 3 - 6 \div 2$ by following the order of operations (PEMDAS/BODMAS). PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right), is our guiding principle in this mathematical journey. First, we'll tackle the parentheses. Inside the parentheses, we have the operation $2 \times 5$, which equals 10. So, we replace $(2 \times 5)$ with 10, simplifying our expression to $4^2 + 10 \times 3 - 6 \div 2$. Next, we address the exponents. We have $4^2$, which means 4 raised to the power of 2, or 4 multiplied by itself. This gives us $4 \times 4 = 16$. Replacing $4^2$ with 16, our expression now becomes $16 + 10 \times 3 - 6 \div 2$. Now, we move on to multiplication and division. These operations are performed from left to right. We first encounter multiplication: $10 \times 3$, which equals 30. Our expression then transforms into $16 + 30 - 6 \div 2$. Following the multiplication, we have division: $6 \div 2$, which equals 3. This simplifies our expression further to $16 + 30 - 3$. Finally, we handle addition and subtraction, again from left to right. We start with addition: $16 + 30$, which equals 46. So, our expression becomes $46 - 3$. Lastly, we perform the subtraction: $46 - 3$, which equals 43. Therefore, the final result of the expression $4^2 + (2 \times 5) \times 3 - 6 \div 2$ is 43. This step-by-step breakdown illustrates the importance of adhering to the order of operations to arrive at the correct solution. Each step builds upon the previous one, ensuring that we maintain accuracy throughout the evaluation process. Understanding and applying PEMDAS is crucial for not only solving mathematical problems but also for developing a logical and systematic approach to problem-solving in general.

Detailed Steps and Calculations

To further clarify the solution, let's break down the evaluation of the expression $4^2 + (2 \times 5) \times 3 - 6 \div 2$ into detailed steps. Understanding each step ensures clarity and reinforces the principles of the order of operations. The first step, according to PEMDAS, is to address the parentheses. Inside the parentheses, we have $2 \times 5$. Multiplying 2 by 5 gives us 10. So, we replace the parentheses with the result: $4^2 + 10 \times 3 - 6 \div 2$. Next, we deal with the exponents. We have $4^2$, which means 4 raised to the power of 2. This is equivalent to 4 multiplied by itself, which equals 16. Substituting this value into the expression, we get: $16 + 10 \times 3 - 6 \div 2$. Now, we move on to multiplication and division. These operations are performed from left to right. First, we encounter multiplication: $10 \times 3$. Multiplying 10 by 3 gives us 30. Our expression now looks like this: $16 + 30 - 6 \div 2$. Following the multiplication, we perform the division: $6 \div 2$. Dividing 6 by 2 results in 3. The expression is further simplified to: $16 + 30 - 3$. Finally, we address addition and subtraction, also from left to right. We start with addition: $16 + 30$. Adding 16 and 30 gives us 46. Our expression is now: $46 - 3$. Lastly, we perform the subtraction: $46 - 3$. Subtracting 3 from 46 gives us 43. Therefore, after meticulously following the order of operations, we find that the value of the expression $4^2 + (2 \times 5) \times 3 - 6 \div 2$ is 43. This detailed breakdown underscores the importance of adhering to each step in the correct sequence to ensure accuracy. Each operation is performed methodically, building upon the previous result to lead us to the final solution. Understanding these steps not only helps in solving this particular expression but also provides a framework for tackling more complex mathematical problems in the future. The consistent application of PEMDAS is the key to success in mathematical evaluations, ensuring that we arrive at the correct answer every time.

Common Mistakes to Avoid

When evaluating expressions, it's crucial to adhere to the order of operations to avoid common mistakes that can lead to incorrect answers. Understanding these pitfalls can significantly improve accuracy and problem-solving skills. A frequent error is neglecting the order of operations altogether. For example, in the expression $4^2 + (2 \times 5) \times 3 - 6 \div 2$, some might be tempted to perform the operations from left to right without considering PEMDAS. This would lead to a completely different and incorrect result. Another common mistake is misunderstanding the precedence between multiplication and division, or addition and subtraction. Remember, while multiplication and division have the same priority, they must be performed from left to right. Similarly, addition and subtraction have equal priority and are also performed from left to right. For instance, if we had an expression like $10 - 4 + 2$, we should subtract 4 from 10 first (resulting in 6) and then add 2, rather than adding 4 and 2 first. A third mistake involves incorrectly handling exponents. Exponents should be evaluated before multiplication, division, addition, and subtraction. Failing to do so can alter the entire calculation. In our example, $4^2$ must be evaluated before any other operation, so we calculate $4 \times 4 = 16$ before proceeding. Another pitfall is mishandling parentheses. Operations within parentheses must be performed first. If there are nested parentheses, the innermost parentheses are addressed first. Ignoring or misinterpreting parentheses can disrupt the entire order of operations and lead to incorrect outcomes. Finally, simple arithmetic errors can also lead to wrong answers. It's crucial to double-check each calculation, especially in complex expressions, to ensure accuracy. Even a small mistake can propagate through the entire problem and result in an incorrect final answer. By being mindful of these common mistakes and consistently applying the order of operations, we can enhance our mathematical skills and ensure that we arrive at the correct solutions. Paying attention to detail and practicing methodical problem-solving are essential for success in mathematics.

Conclusion: The Final Result

In conclusion, by systematically applying the order of operations (PEMDAS), we have successfully evaluated the expression $4^2 + (2 \times 5) \times 3 - 6 \div 2$. Following the established mathematical conventions, we first addressed the parentheses, then the exponents, followed by multiplication and division (from left to right), and finally, addition and subtraction (from left to right). The meticulous step-by-step process began with evaluating the expression within the parentheses, $2 \times 5$, which yielded 10. This simplified the original expression to $4^2 + 10 \times 3 - 6 \div 2$. Next, we tackled the exponent, calculating $4^2$ as $4 \times 4$, which resulted in 16. The expression then became $16 + 10 \times 3 - 6 \div 2$. Moving on to multiplication and division, we first multiplied 10 by 3, obtaining 30, and then divided 6 by 2, which gave us 3. The expression was further simplified to $16 + 30 - 3$. Finally, we performed the addition and subtraction from left to right. Adding 16 and 30 resulted in 46, and subtracting 3 from 46 gave us the final answer. Therefore, the value of the expression $4^2 + (2 \times 5) \times 3 - 6 \div 2$ is 43. This result underscores the importance of adhering to the order of operations in mathematical calculations. Without a consistent and systematic approach, the outcome could easily be incorrect. Understanding and applying PEMDAS ensures that we perform operations in the correct sequence, leading to accurate solutions. This exercise not only demonstrates the practical application of mathematical principles but also reinforces the value of methodical problem-solving in various contexts. The ability to correctly evaluate expressions is a fundamental skill in mathematics, essential for more advanced topics and real-world applications. By mastering the order of operations, we build a strong foundation for future mathematical endeavors and enhance our ability to approach complex problems with confidence and precision.