Simplifying The Expression 3-10+9 X 7 A Step-by-Step Guide

Mathematical expressions, at first glance, might appear daunting with their mix of numbers, operators, and parentheses. However, mastering the art of simplification transforms these complex equations into manageable and solvable problems. This article delves into the step-by-step process of simplifying the expression $3-10+9 imes 7$, providing a clear and concise explanation suitable for learners of all levels. We will explore the fundamental principles of order of operations, unravel the intricacies of arithmetic operations, and ultimately arrive at the correct solution. Whether you're a student grappling with algebraic concepts or simply seeking to enhance your mathematical prowess, this guide offers valuable insights and practical techniques to simplify expressions with confidence.

Understanding the Order of Operations

To simplify any mathematical expression effectively, it's crucial to adhere to the order of operations, often remembered by the acronym PEMDAS or BODMAS. This acronym outlines the sequence in which operations must be performed to ensure accuracy and consistency in the final result. PEMDAS stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). BODMAS, a commonly used alternative, represents 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 underlying principle: a specific order must be followed to simplify mathematical expressions correctly. In our expression, $3-10+9 imes 7$, we have subtraction, addition, and multiplication. According to PEMDAS/BODMAS, multiplication takes precedence over addition and subtraction. This means we must perform the multiplication operation before any other operation in the expression. Neglecting the order of operations can lead to drastically different and incorrect results, highlighting the significance of this fundamental principle in mathematics. By consistently applying the order of operations, we can confidently navigate complex expressions and arrive at accurate solutions, laying a solid foundation for further mathematical exploration.

Step-by-Step Simplification of $3-10+9 imes 7$

Let's apply the order of operations to simplify the expression $3-10+9 imes 7$ step-by-step. Following PEMDAS/BODMAS, the first operation we need to perform is multiplication. In the expression, we have $9 imes 7$, which equals 63. So, we rewrite the expression as $3-10+63$. Now, we are left with addition and subtraction. According to the order of operations, we perform these operations from left to right. First, we perform the subtraction: $3-10$. This results in -7. So, our expression now becomes $-7+63$. Finally, we perform the addition: $-7+63$. This gives us a final result of 56. Therefore, the simplified value of the expression $3-10+9 imes 7$ is 56. This step-by-step breakdown illustrates the importance of adhering to the order of operations to arrive at the correct answer. Each operation is performed in the appropriate sequence, ensuring that the final result accurately reflects the mathematical relationship defined by the expression. By carefully following these steps, we can confidently simplify any mathematical expression, regardless of its complexity.

Detailed Breakdown of Arithmetic Operations

To fully grasp the simplification process, let's delve into the arithmetic operations involved in the expression $3-10+9 imes 7$. The expression involves three primary operations: subtraction, addition, and multiplication. We've already established that multiplication takes precedence. So, let's first focus on $9 imes 7$. Multiplication is the process of repeated addition. In this case, $9 imes 7$ means adding 9 to itself 7 times, or adding 7 to itself 9 times, which equals 63. Next, we address the subtraction: $3-10$. Subtraction can be thought of as adding the negative of a number. So, $3-10$ is the same as $3 + (-10)$. When adding numbers with different signs, we find the difference between their absolute values and assign the sign of the number with the larger absolute value. The absolute value of 3 is 3, and the absolute value of -10 is 10. The difference between 10 and 3 is 7. Since -10 has a larger absolute value, the result is -7. Finally, we perform the addition: $-7+63$. This involves adding a negative number to a positive number. Again, we find the difference between their absolute values. The absolute value of -7 is 7, and the absolute value of 63 is 63. The difference between 63 and 7 is 56. Since 63 is positive and has a larger absolute value, the result is positive 56. This detailed breakdown highlights the fundamental principles of arithmetic operations, emphasizing how each operation contributes to the final solution. By understanding these operations at a granular level, we can confidently tackle more complex mathematical problems.

Common Mistakes to Avoid When Simplifying Expressions

Simplifying mathematical expressions can be tricky, and certain common mistakes can lead to incorrect answers. One of the most frequent errors is disregarding the order of operations. As we've emphasized, PEMDAS/BODMAS is crucial. Failing to prioritize multiplication and division before addition and subtraction, or neglecting parentheses, can result in a drastically different outcome. For instance, if we were to incorrectly perform the addition in $3-10+9 imes 7$ before the multiplication, we would get $3-19 imes 7$, which is clearly wrong. Another common mistake involves mishandling negative signs. When subtracting a number, especially a larger one, from a smaller number, the result will be negative. It's essential to pay close attention to the signs of numbers and apply the rules of addition and subtraction for signed numbers correctly. A third mistake is making arithmetic errors in basic calculations. Even if the order of operations is followed correctly, a simple addition or multiplication mistake can throw off the entire solution. It's always wise to double-check calculations, especially in complex expressions. Finally, some individuals might try to simplify expressions from left to right without considering the order of operations. This approach is almost guaranteed to produce incorrect results. By being aware of these common pitfalls and practicing mindful calculation, we can significantly reduce the likelihood of errors and improve our accuracy in simplifying expressions. Consistent attention to detail and a thorough understanding of mathematical principles are key to success.

Practice Problems and Solutions

To solidify your understanding of simplifying expressions, let's work through a few practice problems. These examples will demonstrate the application of the order of operations and highlight the techniques we've discussed.

Problem 1: Simplify $12 + 4 imes (7 - 2)$

Solution: First, we address the parentheses: $7 - 2 = 5$. So, the expression becomes $12 + 4 imes 5$. Next, we perform the multiplication: $4 imes 5 = 20$. Now we have $12 + 20$, which equals 32. Therefore, the simplified value of $12 + 4 imes (7 - 2)$ is 32.

Problem 2: Simplify $18 ext{ ÷ } 3 + 2 imes 4 - 1$

Solution: We begin with division and multiplication from left to right. First, $18 ext{ ÷ } 3 = 6$. The expression becomes $6 + 2 imes 4 - 1$. Next, $2 imes 4 = 8$. Now we have $6 + 8 - 1$. We perform addition and subtraction from left to right. $6 + 8 = 14$, and then $14 - 1 = 13$. So, the simplified value of $18 ext{ ÷ } 3 + 2 imes 4 - 1$ is 13.

Problem 3: Simplify $5 imes (10 - 2) + 15 ext{ ÷ } 3$

Solution: First, we address the parentheses: $10 - 2 = 8$. The expression becomes $5 imes 8 + 15 ext{ ÷ } 3$. Next, we perform multiplication and division from left to right. $5 imes 8 = 40$, and $15 ext{ ÷ } 3 = 5$. Now we have $40 + 5$, which equals 45. Therefore, the simplified value of $5 imes (10 - 2) + 15 ext{ ÷ } 3$ is 45.

By working through these practice problems, you can reinforce your understanding of simplifying expressions and develop the skills needed to tackle more challenging problems. Remember to always adhere to the order of operations and pay close attention to detail.

Conclusion

In conclusion, simplifying mathematical expressions is a fundamental skill that requires a solid understanding of the order of operations and arithmetic principles. By consistently applying PEMDAS/BODMAS, we can navigate complex expressions with confidence and arrive at accurate solutions. This article has provided a comprehensive guide to simplifying the expression $3-10+9 imes 7$, including a detailed step-by-step breakdown, explanations of arithmetic operations, and common mistakes to avoid. Through practice problems and their solutions, we've reinforced the key concepts and techniques involved in simplifying expressions. Mastering these skills not only enhances our mathematical abilities but also lays a strong foundation for tackling more advanced mathematical concepts. So, embrace the principles discussed, practice diligently, and you'll be well-equipped to simplify any mathematical expression that comes your way. Remember, mathematics is a journey of continuous learning and practice, and each step you take brings you closer to mastering this fascinating subject.