Simplify The Expression Using Order Of Operations

by ADMIN 50 views

In the realm of mathematics, simplifying expressions is a fundamental skill. It allows us to break down complex equations into manageable components, ultimately leading to a clear and concise solution. One of the most crucial aspects of simplifying expressions is understanding and applying the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This guide will delve into the simplification of a specific expression,

9 \cdot 3 - 6 \cdot 5 - 4 \cdot 8 = \square

using the order of operations as our guiding principle. We will dissect each step, providing clear explanations and insights to ensure a comprehensive understanding of the process.

Understanding the Order of Operations (PEMDAS/BODMAS)

Before we embark on simplifying the expression, it's imperative to grasp the order of operations, a set of rules that dictates the sequence in which mathematical operations should be performed. The acronym PEMDAS (or BODMAS, which stands for Brackets, Orders, Division and Multiplication, Addition and Subtraction) serves as a handy mnemonic device to remember this order:

  • Parentheses (or Brackets): Operations enclosed within parentheses or brackets are always performed first.
  • Exponents (or Orders): Next, we address exponents or orders, such as squares, cubes, and other powers.
  • Multiplication and Division: Multiplication and division are performed from left to right.
  • Addition and Subtraction: Finally, addition and subtraction are carried out from left to right.

This order is not arbitrary; it's a convention established to ensure consistency and avoid ambiguity in mathematical calculations. Without a standardized order, the same expression could yield different results depending on the sequence of operations performed. For instance, consider the expression 2 + 3 * 4. If we perform addition first, we get 5 * 4 = 20. However, if we follow the order of operations and perform multiplication first, we get 2 + 12 = 14. The latter is the correct answer, highlighting the importance of adhering to PEMDAS/BODMAS.

In our expression, 9 \cdot 3 - 6 \cdot 5 - 4 \cdot 8, we have multiplication and subtraction operations. According to PEMDAS, multiplication takes precedence over subtraction. Therefore, we will first perform all the multiplications before moving on to the subtractions. This methodical approach will guarantee that we arrive at the correct simplified answer.

Step 1: Performing the Multiplications

The expression 9 \cdot 3 - 6 \cdot 5 - 4 \cdot 8 contains three multiplication operations: 9 \cdot 3, 6 \cdot 5, and 4 \cdot 8. We will perform each of these multiplications individually:

  • 9 \cdot 3 = 27
  • 6 \cdot 5 = 30
  • 4 \cdot 8 = 32

By carrying out these multiplications, we transform the original expression into a simpler form: 27 - 30 - 32. This step is crucial as it reduces the complexity of the expression, making it easier to manage. We have effectively eliminated the multiplication operations, paving the way for the next step in the simplification process.

It's important to note that we perform the multiplications from left to right, as dictated by the order of operations. This ensures that we maintain the correct sequence and arrive at the accurate result. In cases where both multiplication and division are present, we would similarly proceed from left to right. The same principle applies to addition and subtraction.

Now that we have completed the multiplications, we are left with a series of subtraction operations. In the next step, we will tackle these subtractions, again adhering to the order of operations and proceeding from left to right. This systematic approach guarantees that we accurately simplify the expression and arrive at the final answer.

Step 2: Performing the Subtractions

After performing the multiplications, our expression has been simplified to 27 - 30 - 32. Now, we need to execute the subtraction operations. According to the order of operations, subtraction and addition are performed from left to right. Therefore, we will first subtract 30 from 27, and then subtract 32 from the result.

Let's break it down step-by-step:

  1. 27 - 30 = -3

    Subtracting 30 from 27 yields a negative result, -3. This is because we are subtracting a larger number from a smaller number. It's important to be comfortable working with negative numbers when simplifying expressions, as they often arise in mathematical calculations.

  2. -3 - 32 = -35

    Next, we subtract 32 from -3. This is equivalent to adding -32 to -3, resulting in -35. When subtracting a positive number from a negative number, the result becomes more negative.

By performing these subtractions sequentially from left to right, we have successfully simplified the expression. The final result is -35. This process highlights the importance of following the order of operations to ensure accuracy. Had we performed the subtractions in a different order, we would have arrived at an incorrect answer.

Therefore, the simplified form of the expression 9 \cdot 3 - 6 \cdot 5 - 4 \cdot 8 is -35. We have systematically applied the order of operations, performing the multiplications first and then the subtractions, to arrive at this final answer. This meticulous approach is crucial for simplifying any mathematical expression, regardless of its complexity.

Final Answer

In conclusion, by meticulously following the order of operations, we have successfully simplified the expression 9 \cdot 3 - 6 \cdot 5 - 4 \cdot 8. The steps involved performing the multiplications first, followed by the subtractions from left to right. This process led us to the final answer:

9 \cdot 3 - 6 \cdot 5 - 4 \cdot 8 = -35

This exercise underscores the significance of adhering to the rules of PEMDAS/BODMAS to ensure accuracy in mathematical calculations. By understanding and applying the order of operations, we can confidently tackle complex expressions and arrive at the correct solutions. Remember, mathematics is a language, and the order of operations is a crucial grammatical rule that enables us to communicate mathematical ideas effectively and without ambiguity. So, embrace the power of PEMDAS/BODMAS, and you'll find that simplifying expressions becomes a much more straightforward and rewarding endeavor.

This comprehensive guide has not only provided the solution to the given expression but has also illuminated the fundamental principles of order of operations. This knowledge will serve as a valuable foundation for tackling more advanced mathematical concepts in the future. Keep practicing, and you'll master the art of simplifying expressions with ease!