Simplifying The Expression [(3-8)^2+2] ÷ (9-6) A Step-by-Step Guide

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This article delves into the process of simplifying the mathematical expression [(3-8)^2+2] ÷ (9-6), providing a step-by-step solution and explanation. We will break down the expression, applying the order of operations (PEMDAS/BODMAS) to arrive at the correct answer. This guide aims to not only provide the solution but also to enhance your understanding of mathematical simplification, making it easier to tackle similar problems in the future. By the end of this article, you will be equipped with the knowledge and skills to confidently simplify complex mathematical expressions.

Understanding the Order of Operations (PEMDAS/BODMAS)

Before we dive into the simplification, it's crucial to understand the order of operations, often remembered by the acronyms PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) or BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction). This order dictates the sequence in which operations must be performed to ensure the correct result. Ignoring this order can lead to incorrect answers, so it’s essential to adhere to it strictly.

  • Parentheses/Brackets: Operations within parentheses or brackets are performed first. This step takes precedence over all other operations.
  • Exponents/Orders: Next, we handle exponents or orders (powers and square roots, etc.). These operations are performed after the parentheses/brackets are simplified.
  • Multiplication and Division: Multiplication and division are performed from left to right. These operations have equal precedence, so we perform them in the order they appear.
  • Addition and Subtraction: Finally, addition and subtraction are performed from left to right. Similar to multiplication and division, these operations have equal precedence and are performed in the order they appear.

By consistently applying the order of operations, you can simplify complex expressions accurately and efficiently. This foundational principle is the key to solving mathematical problems with confidence.

Step-by-Step Simplification of [(3-8)^2+2] ÷ (9-6)

Now, let's apply the PEMDAS/BODMAS rule to simplify the given expression [(3-8)^2+2] ÷ (9-6) step by step. Each step will be explained in detail to ensure a clear understanding of the process.

1. Simplify within the Parentheses/Brackets

The expression contains two sets of parentheses: (3-8) and (9-6). We'll start by simplifying these.

  • (3-8) = -5: Subtracting 8 from 3 gives us -5.
  • (9-6) = 3: Subtracting 6 from 9 gives us 3.

Now, our expression becomes: [(-5)^2+2] ÷ 3

2. Evaluate the Exponent

Next, we need to evaluate the exponent, which is (-5)^2. Remember that squaring a negative number results in a positive number.

  • (-5)^2 = (-5) * (-5) = 25

Our expression now looks like this: [25+2] ÷ 3

3. Simplify the Remaining Parentheses/Brackets

We still have brackets containing the expression 25+2. Let's simplify this.

  • 25 + 2 = 27

The expression is now simplified to: 27 ÷ 3

4. Perform the Division

Finally, we perform the division operation.

  • 27 ÷ 3 = 9

Therefore, the simplified value of the expression [(3-8)^2+2] ÷ (9-6) is 9.

Detailed Breakdown of Each Step

To further solidify your understanding, let’s delve deeper into each step of the simplification process. This detailed breakdown will help clarify any potential confusion and reinforce the application of PEMDAS/BODMAS.

1. Simplifying within Parentheses: A Closer Look

The first step in simplifying the expression involves addressing the operations within the parentheses. Parentheses serve as grouping symbols, indicating that the enclosed operations should be performed before any operations outside the parentheses. In our expression, we have two sets of parentheses: (3-8) and (9-6). Simplifying these is a straightforward application of subtraction.

  • (3-8) = -5: This calculation involves subtracting a larger number (8) from a smaller number (3), resulting in a negative value. Understanding how to work with negative numbers is crucial in mathematics, and this step provides a practical example. The result, -5, is then carried forward to the next step.
  • (9-6) = 3: This is a simpler subtraction, where we subtract 6 from 9, yielding a positive result of 3. This result will be used as the divisor in the final division step.

By simplifying the expressions within the parentheses first, we adhere to the PEMDAS/BODMAS rule and set the stage for the subsequent operations.

2. Evaluating the Exponent: Understanding Powers

After simplifying the parentheses, the next step is to evaluate the exponent. In our expression, we have (-5)^2, which means -5 raised to the power of 2. This is equivalent to multiplying -5 by itself.

  • (-5)^2 = (-5) * (-5) = 25: This calculation demonstrates an important rule: a negative number multiplied by another negative number results in a positive number. The result of this operation is 25, which replaces (-5)^2 in the expression. Understanding exponents and their properties is essential for simplifying expressions and solving equations.

3. Simplifying the Remaining Brackets: Maintaining Order

At this stage, we have simplified the expression inside the square brackets to [25+2]. Although these are brackets rather than parentheses, they still serve as grouping symbols, indicating that the addition should be performed before the division. This step reinforces the importance of maintaining the correct order of operations.

  • 25 + 2 = 27: This simple addition combines the results of the exponent evaluation and sets the stage for the final division step. The expression within the brackets is now simplified to a single number, 27.

4. Performing the Division: The Final Step

The final step in simplifying the expression is to perform the division. We now have the simplified expression 27 ÷ 3. This is a straightforward division operation.

  • 27 ÷ 3 = 9: Dividing 27 by 3 gives us the final answer of 9. This completes the simplification process, demonstrating how each step contributes to the overall solution.

Conclusion: The Correct Answer and Key Takeaways

In conclusion, the simplified value of the expression [(3-8)^2+2] ÷ (9-6) is 9. Therefore, the correct answer is B. 9. This problem highlights the importance of following the order of operations (PEMDAS/BODMAS) and understanding the rules of arithmetic. By breaking down the expression into smaller, manageable steps, we can simplify complex problems with ease and accuracy.

The key takeaways from this exercise are:

  • Mastering the Order of Operations: PEMDAS/BODMAS is the foundation of mathematical simplification. Always follow this order to ensure correct results.
  • Understanding Parentheses/Brackets: These symbols indicate which operations to perform first, grouping expressions together.
  • Working with Exponents: Remember that exponents represent repeated multiplication, and negative numbers raised to an even power become positive.
  • Step-by-Step Approach: Breaking down complex expressions into smaller steps makes the simplification process more manageable and less prone to errors.

By applying these principles, you can confidently tackle a wide range of mathematical simplification problems. Practice is key to mastering these skills, so continue to work through various examples to solidify your understanding.

This comprehensive guide has provided a detailed explanation of how to simplify the expression [(3-8)^2+2] ÷ (9-6). We hope this article has been helpful in enhancing your mathematical skills and understanding. Remember, the key to success in mathematics is a combination of knowledge, practice, and a systematic approach to problem-solving.