How Do You Solve The Mathematical Expression {49÷(-7) -(-15)} + 4 × (-7+3) And Get The Answer?
Mathematics, often perceived as a realm of complex equations and abstract concepts, is fundamentally a tool for problem-solving and understanding the world around us. One of the cornerstones of mathematical proficiency is the ability to accurately evaluate expressions, navigating the order of operations and the intricacies of signed numbers. In this article, we will embark on a journey to dissect the expression {49÷(-7) -(-15)} + 4 × (-7+3), meticulously unraveling each step to illuminate the path towards the correct solution. The expression may seem daunting at first glance, but with a systematic approach and a clear understanding of the underlying principles, we can confidently conquer it. Our mission is not merely to arrive at the final answer but also to gain a deeper appreciation for the logical flow and precision that mathematics demands. By carefully examining each component of the expression, we will reinforce our grasp of arithmetic operations, signed numbers, and the crucial order of operations. This journey of mathematical exploration will not only empower us to solve similar problems but also cultivate a stronger foundation for more advanced mathematical concepts. So, let us delve into the world of numbers, symbols, and equations, and together, we will unravel the mystery behind this mathematical expression.
Understanding the Order of Operations: PEMDAS/BODMAS
Before we embark on solving the expression, it's crucial to grasp the order of operations, often remembered by the acronyms PEMDAS or BODMAS. This mnemonic device provides a roadmap for evaluating expressions, ensuring that we perform operations in the correct sequence. PEMDAS stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). BODMAS is a similar acronym, representing Brackets, Orders (exponents and roots), Division and Multiplication (from left to right), and Addition and Subtraction (from left to right). Both acronyms convey the same fundamental principle: certain operations take precedence over others. Parentheses (or brackets) are always addressed first, as they group terms and dictate the order in which they should be evaluated. Next come exponents or orders, representing powers and roots. Multiplication and division hold equal priority and are performed from left to right, as they appear in the expression. Similarly, addition and subtraction share the same level of precedence and are carried out from left to right. By adhering to this established order, we eliminate ambiguity and ensure that we arrive at the correct solution. The order of operations acts as a universal language in mathematics, providing a consistent framework for interpreting and evaluating expressions. Without it, mathematical expressions could be interpreted in multiple ways, leading to conflicting results. The PEMDAS/BODMAS principle is not just a set of rules; it's a cornerstone of mathematical logic, ensuring clarity and consistency in our calculations. As we delve into the expression {49÷(-7) -(-15)} + 4 × (-7+3), we will constantly refer back to the order of operations, using it as our guide to navigate the complexities of the expression. By mastering the order of operations, we not only solve individual problems but also cultivate a deeper understanding of the mathematical structure and the logical principles that govern it.
Step-by-Step Solution
Now, let's methodically solve the expression 49÷(-7) -(-15)} + 4 × (-7+3)**, breaking it down into manageable steps to ensure clarity and accuracy. Our journey begins within the curly braces, where we encounter a combination of division and subtraction. According to the order of operations, we tackle the division first. 49 divided by -7 yields -7. So, we rewrite the expression as {-7 -(-15)} + 4 × (-7+3). Next, we address the subtraction within the braces. Subtracting a negative number is equivalent to adding its positive counterpart. Therefore, -7 - (-15) becomes -7 + 15, which equals 8. Our expression now simplifies to 8 + 4 × (-7+3). Moving forward, we encounter a multiplication operation. Before we can multiply, we need to resolve the expression within the parentheses + 4 × (-7+3) is -8. Each step in this process was carefully executed, adhering to the order of operations and the rules of signed numbers. By breaking down the complex expression into smaller, more manageable parts, we were able to navigate the intricacies and arrive at the correct answer. This step-by-step approach is not only essential for solving mathematical problems but also for developing a deeper understanding of the underlying concepts. As we move forward in our mathematical journey, we will continue to employ this systematic approach, tackling challenges with confidence and precision.
1. Solve the Parentheses: (-7 + 3)
The initial step in solving our mathematical puzzle, {49÷(-7) -(-15)} + 4 × (-7+3), involves addressing the parentheses. According to the order of operations, expressions enclosed within parentheses take precedence over other operations. In this case, we have (-7 + 3). This simple addition of two signed numbers requires us to combine a negative number (-7) with a positive number (3). To perform this operation, we can visualize a number line. Starting at -7, we move 3 units to the right, which brings us to -4. Alternatively, we can consider the absolute values of the numbers. The absolute value of -7 is 7, and the absolute value of 3 is 3. Since the numbers have different signs, we subtract the smaller absolute value from the larger one: 7 - 3 = 4. We then assign the sign of the number with the larger absolute value, which in this case is -7, making the result -4. Therefore, (-7 + 3) equals -4. This seemingly small step is crucial as it simplifies the expression and paves the way for subsequent calculations. By resolving the parentheses first, we adhere to the order of operations and ensure that our final answer is accurate. The result of this step, -4, will be used in the next stage of our calculation, highlighting the interconnectedness of mathematical operations. The ability to confidently handle signed number addition is a fundamental skill in mathematics, and this step provides an excellent opportunity to reinforce that understanding. As we continue to unravel the expression, we will appreciate how each step builds upon the previous one, guiding us towards the ultimate solution.
2. Division within the Braces: 49 ÷ (-7)
Following our methodical approach, we now turn our attention to the next operation within the curly braces: 49 ÷ (-7). This division involves a positive number (49) and a negative number (-7). The rules of signed number division dictate that when we divide a positive number by a negative number, the result is a negative number. To perform the division, we first consider the absolute values of the numbers. The absolute value of 49 is 49, and the absolute value of -7 is 7. Dividing 49 by 7 gives us 7. However, since we are dividing a positive number by a negative number, the result is -7. Therefore, 49 ÷ (-7) equals -7. This step is another crucial building block in our journey towards solving the entire expression. By accurately performing the division, we simplify the expression further and bring ourselves closer to the final answer. The concept of signed number division is a cornerstone of arithmetic, and mastering it is essential for success in higher-level mathematics. This step provides a practical application of the rules of signed number division, reinforcing our understanding and solidifying our skills. As we continue our exploration, we will see how this result, -7, interacts with other terms in the expression, ultimately contributing to the final solution. The attention to detail in each step, including the correct application of signed number rules, is what ensures the accuracy of our calculations. Mathematical problem-solving is not just about arriving at the answer; it's about the process and the understanding we gain along the way.
3. Handling the Double Negative: -(-15)
As we progress through the expression {49÷(-7) -(-15)} + 4 × (-7+3), we encounter a situation that often causes confusion: a double negative. Specifically, we have -(-15). In mathematics, a double negative is equivalent to a positive. This concept stems from the fundamental properties of number systems and the way negative numbers interact with subtraction. To understand why this is the case, we can think of subtraction as adding the opposite. So, -(-15) can be interpreted as subtracting -15, which is the same as adding the opposite of -15. The opposite of -15 is 15, so -(-15) is equivalent to adding 15. Mathematically, this can be represented as -(-15) = +15. This transformation is crucial for simplifying the expression and ensuring that we arrive at the correct solution. Failing to recognize and correctly handle the double negative can lead to errors in subsequent calculations and ultimately an incorrect final answer. By understanding the underlying principle of double negatives, we gain a deeper appreciation for the structure and logic of mathematical operations. This step highlights the importance of paying close attention to signs and symbols in mathematical expressions. A seemingly small detail, like a double negative, can have a significant impact on the final result. As we continue to solve the expression, we will see how this step, the conversion of -(-15) to +15, contributes to the overall simplification and leads us closer to the accurate solution. Mathematical proficiency is not just about performing calculations; it's about understanding the rules and applying them with precision.
4. Evaluate within the Braces: -7 - (-15)
Having tackled the division and the double negative within the curly braces, we now focus on evaluating the remaining expression within the braces: -7 - (-15). This operation combines the results of our previous steps, requiring us to subtract a negative number from another negative number. As we established earlier, subtracting a negative number is equivalent to adding its positive counterpart. Therefore, -7 - (-15) can be rewritten as -7 + 15. Now we have a simple addition of two numbers with different signs. To perform this addition, we consider the absolute values of the numbers. The absolute value of -7 is 7, and the absolute value of 15 is 15. Since the numbers have different signs, we subtract the smaller absolute value from the larger one: 15 - 7 = 8. We then assign the sign of the number with the larger absolute value, which in this case is 15, making the result positive. Therefore, -7 - (-15) equals 8. This step further simplifies our expression, bringing us closer to the final solution. By accurately evaluating the expression within the braces, we eliminate a significant source of complexity and pave the way for the remaining operations. The ability to confidently handle signed number operations is a fundamental skill in mathematics, and this step provides an excellent opportunity to reinforce that understanding. As we continue to unravel the expression, we will see how this result, 8, interacts with the remaining terms, ultimately contributing to the final solution. The systematic approach we have adopted, breaking down the complex expression into smaller, more manageable steps, is a key to success in mathematical problem-solving.
5. Multiplication: 4 × (-4)
With the expression within the curly braces fully evaluated, we now shift our focus to the remaining operations outside the braces. Our expression currently stands as 8 + 4 × (-4). According to the order of operations, multiplication takes precedence over addition. Therefore, we must perform the multiplication operation before we can add. We have 4 multiplied by -4. This involves multiplying a positive number by a negative number. The rules of signed number multiplication dictate that when we multiply a positive number by a negative number, the result is a negative number. To perform the multiplication, we first consider the absolute values of the numbers. The absolute value of 4 is 4, and the absolute value of -4 is 4. Multiplying 4 by 4 gives us 16. However, since we are multiplying a positive number by a negative number, the result is -16. Therefore, 4 × (-4) equals -16. This step significantly simplifies our expression, reducing it to a simple addition problem. By accurately performing the multiplication, we eliminate a major source of complexity and bring ourselves closer to the final answer. The concept of signed number multiplication is a cornerstone of arithmetic, and mastering it is essential for success in higher-level mathematics. This step provides a practical application of the rules of signed number multiplication, reinforcing our understanding and solidifying our skills. As we move to the final step, the addition, we will see how this result, -16, interacts with the remaining term, 8, to produce the ultimate solution. Mathematical precision is paramount, and the careful application of signed number rules is crucial for achieving accurate results.
6. Final Addition: 8 + (-16)
The culmination of our step-by-step journey brings us to the final operation: the addition of 8 and -16. Our expression has been meticulously simplified to this point, and now we must perform this final calculation to arrive at the solution. We are adding a positive number (8) to a negative number (-16). To perform this operation, we can visualize a number line. Starting at 8, we move 16 units to the left, which brings us to -8. Alternatively, we can consider the absolute values of the numbers. The absolute value of 8 is 8, and the absolute value of -16 is 16. Since the numbers have different signs, we subtract the smaller absolute value from the larger one: 16 - 8 = 8. We then assign the sign of the number with the larger absolute value, which in this case is -16, making the result negative. Therefore, 8 + (-16) equals -8. This final step completes our journey of unraveling the expression {49÷(-7) -(-15)} + 4 × (-7+3). By meticulously following the order of operations and carefully applying the rules of signed number arithmetic, we have successfully navigated the complexities of the expression and arrived at the correct answer. The solution is -8. This process highlights the importance of a systematic approach to problem-solving in mathematics. By breaking down a complex problem into smaller, more manageable steps, we can conquer even the most challenging expressions. The sense of accomplishment that comes from successfully solving a mathematical problem is not just about the answer; it's about the understanding and skills we develop along the way. As we continue our mathematical journey, we will carry with us the lessons learned from this experience, empowering us to tackle future challenges with confidence and precision.
Common Mistakes and How to Avoid Them
Navigating the realm of mathematical expressions can be fraught with potential pitfalls, and it's crucial to be aware of common mistakes to avoid them. One frequent error is disregarding the order of operations. Failing to adhere to PEMDAS/BODMAS can lead to incorrect results. For instance, in the expression {49÷(-7) -(-15)} + 4 × (-7+3), if we were to add 8 and 4 before performing the multiplication, we would deviate from the correct path and arrive at an erroneous solution. To avoid this, always remember to prioritize parentheses, exponents, multiplication and division (from left to right), and finally addition and subtraction (from left to right). Another common mistake lies in mishandling signed numbers. The rules of signed number arithmetic can be tricky, especially when dealing with subtraction and double negatives. Confusing the rules for addition, subtraction, multiplication, and division of signed numbers can lead to significant errors. A clear understanding of these rules is essential. For example, remember that subtracting a negative number is the same as adding its positive counterpart, and multiplying or dividing numbers with different signs results in a negative answer. It's also important to pay close attention to signs throughout the calculation. A misplaced negative sign can throw off the entire solution. Double-check each step to ensure that the signs are correctly applied. Another potential pitfall is rushing through the problem. Mathematical precision requires patience and attention to detail. Taking the time to carefully evaluate each step, breaking down the expression into smaller, more manageable parts, can significantly reduce the likelihood of errors. Finally, practice makes perfect. The more you work with mathematical expressions, the more comfortable and confident you will become in applying the rules and avoiding common mistakes. Regularly solving problems and reviewing your work can help solidify your understanding and improve your accuracy. By being aware of these common mistakes and actively working to avoid them, you can enhance your mathematical problem-solving skills and achieve greater success.
Conclusion
In conclusion, the journey of solving the expression {49÷(-7) -(-15)} + 4 × (-7+3) has been a valuable exercise in mathematical problem-solving. We have meticulously dissected each step, reinforcing our understanding of the order of operations, signed number arithmetic, and the importance of precision in calculations. The final solution, -8, is not just a number; it represents the culmination of a logical process, a testament to the power of mathematical reasoning. We began by establishing the fundamental principle of the order of operations, PEMDAS/BODMAS, which served as our guiding framework throughout the process. We then systematically addressed each component of the expression, starting with the parentheses, progressing through division, subtraction, multiplication, and finally arriving at the addition. Along the way, we paid close attention to the rules of signed numbers, ensuring that we accurately handled positive and negative values. We also highlighted common mistakes that can occur when evaluating expressions and provided strategies for avoiding them. By understanding these potential pitfalls, we can become more mindful and accurate in our calculations. The ability to solve mathematical expressions is not just an academic skill; it's a valuable tool that can be applied in various aspects of life, from budgeting and finance to science and engineering. The logical thinking and problem-solving skills honed through mathematics are transferable to many other domains. As we conclude this exploration, we encourage you to continue practicing and applying these principles. The more you engage with mathematical problems, the more confident and proficient you will become. Mathematics is not just a subject to be studied; it's a language to be learned, a tool to be wielded, and a world to be explored. Embrace the challenge, celebrate the successes, and continue on your mathematical journey with curiosity and enthusiasm.