Simplifying Mathematical Expressions A Step By Step Guide For 4+(-3)-2 ×(-6)

Jul 11, 2025 by Admin 77 views

Understanding the Order of Operations

In mathematics, simplifying expressions requires a clear understanding of the order of operations. Often remembered by the acronym PEMDAS, this principle dictates the sequence in which calculations should be performed. PEMDAS stands for:

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

When simplifying the expression $4+(-3)-2 \times(-6)$ , it’s crucial to follow PEMDAS meticulously to arrive at the correct answer. Initially, we observe that there are no exponents, but parentheses and multiplication are present. These operations need to be addressed in the correct order. By adhering to PEMDAS, we ensure accuracy and avoid common pitfalls that can arise from performing operations in the wrong sequence. Math is sequential, and understanding PEMDAS helps in correctly deciphering and solving complex expressions. This methodical approach is not just a rule but a fundamental aspect of mathematical logic, ensuring consistency and precision in problem-solving. Without a clear order, mathematical expressions could lead to multiple interpretations and results, thus highlighting the importance of PEMDAS in maintaining mathematical integrity.

Step-by-Step Simplification of the Expression

To simplify the expression $4+(-3)-2 \times(-6)$ , we will break it down step-by-step, meticulously following the order of operations (PEMDAS). The first element to address is the term within the parentheses. We have $4 + (-3)$ , which involves adding a negative number. Adding a negative number is equivalent to subtraction, so $4 + (-3)$ simplifies to $4 - 3$ , which equals 1. This step sets the foundation for the rest of the calculation by reducing part of the expression to a simpler form.

Next, we turn our attention to the multiplication operation. According to PEMDAS, multiplication should be performed before addition and subtraction. In the expression, we have $-2 \times (-6)$ . When multiplying two negative numbers, the result is a positive number. Therefore, $-2 \times (-6)$ equals 12. This multiplication step is a critical component of simplifying the entire expression. The sign of the result is as important as the numerical value, as it will affect the subsequent operations.

With the multiplication completed, our expression now looks like $1 + 12$ . The remaining operation is addition. Adding 1 and 12 gives us 13. Thus, the final simplified value of the expression $4+(-3)-2 \times(-6)$ is 13. This step-by-step approach not only provides the solution but also illustrates the importance of handling each operation in the correct sequence. By methodically working through the expression, we avoid errors and arrive at the accurate final answer.

Detailed Breakdown: Parentheses and Initial Addition

The initial phase in simplifying the expression $4+(-3)-2 \times(-6)$ focuses on the parentheses and the immediate addition. The expression contains the term $4 + (-3)$ , which requires careful attention to the rules of adding integers. When adding a negative number to a positive number, it's equivalent to subtracting the absolute value of the negative number from the positive number. Therefore, $4 + (-3)$ is the same as $4 - 3$ .

Performing this subtraction, we find that $4 - 3$ equals 1. This step simplifies the expression, reducing it to $1 - 2 \times (-6)$ . This initial simplification is crucial because it reduces the complexity of the expression and sets the stage for the next operations. Dealing with parentheses and additions involving negative numbers correctly is essential for mastering algebraic simplification. By accurately handling this step, we ensure that the subsequent operations are performed on the correct values, leading to a correct final answer. The process highlights how each individual operation contributes to the overall simplification and emphasizes the importance of precision in arithmetic.

Multiplication of Negative Numbers

The next critical step in simplifying the expression $4+(-3)-2 \times(-6)$ involves handling the multiplication of negative numbers. Specifically, we need to address the term $-2 \times (-6)$ . One of the fundamental rules of arithmetic is that the product of two negative numbers is a positive number. This rule is crucial for accurately simplifying expressions involving negative signs. When we multiply $-2$ by $-6$ , we are essentially finding the result of a negative times a negative, which will yield a positive product.

In this case, $-2 \times (-6)$ equals 12. This conversion of two negatives into a positive is a key aspect of the calculation. The positive 12 significantly impacts the subsequent steps, ensuring the expression is correctly simplified. Understanding and correctly applying the rules of multiplying negative numbers is essential in algebra and arithmetic. It helps avoid common mistakes that arise from sign errors and leads to accurate solutions. This specific multiplication demonstrates the principle that the sign of the numbers involved is as important as their numerical values, particularly in more complex mathematical problems. By carefully addressing the multiplication of negative numbers, we maintain the integrity of the calculation and pave the way for the final steps in simplifying the expression.

Final Steps: Subtraction and the Final Result

After addressing the parentheses and multiplication in the expression $4+(-3)-2 \times(-6)$ , we arrive at the final steps of simplification, which involve subtraction. The expression has now been reduced to $1 + 12$ . This final stage requires us to perform the addition to obtain the ultimate result. Adding 1 and 12 is a straightforward arithmetic operation. When we combine these two numbers, we get:

$1 + 12 = 13$

Therefore, the simplified value of the original expression $4+(-3)-2 \times(-6)$ is 13. This final step demonstrates how all the previous operations culminate in a single, concise answer. Each step in the process, from handling parentheses and negative numbers to performing multiplication and addition, is crucial for arriving at the correct solution. The accuracy and precision applied in each stage ensure the final result is valid. This complete simplification process illustrates the importance of following the order of operations and paying attention to detail in mathematical calculations. The final answer, 13, represents the culmination of these efforts, showcasing the power of methodical mathematical problem-solving.

Conclusion

In conclusion, simplifying the expression $4+(-3)-2 \times(-6)$ involves a series of methodical steps following the order of operations (PEMDAS). We begin by addressing the parentheses, simplifying $4 + (-3)$ to 1. Then, we tackle the multiplication, where $-2 \times (-6)$ equals 12. Finally, we perform the addition, $1 + 12$ , which yields the final result of 13. Each step in this process is crucial, and correctly executing each operation ensures an accurate simplification.

The importance of understanding and applying the order of operations cannot be overstated. PEMDAS provides a clear guideline for how to approach mathematical expressions, ensuring consistency and accuracy in the results. Whether dealing with simple arithmetic or complex algebraic equations, the principles of PEMDAS remain fundamental. By following these principles, we avoid common errors and achieve correct solutions.

Furthermore, this exercise demonstrates the significance of attention to detail in mathematics. Each sign, each operation, and each step must be carefully considered to avoid mistakes. The meticulous approach not only leads to the correct answer but also reinforces mathematical reasoning and problem-solving skills. Simplifying expressions like this one enhances our ability to tackle more complex mathematical challenges and fosters a deeper understanding of mathematical concepts. This comprehensive approach to simplification is essential for both academic and practical applications of mathematics.