Simplify (7+1)^2-(11+3^2) ÷ 4: A Step-by-Step Guide

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In this article, we will walk through the process of simplifying the mathematical expression extbf{(7+1)2-(11+32) ÷ 4}. This involves following the order of operations, commonly remembered by the acronym PEMDAS/BODMAS (Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction). Understanding and correctly applying this order is crucial to solving mathematical expressions accurately. We will break down each step, providing a clear and concise explanation to help you grasp the underlying concepts. By the end of this article, you will not only be able to solve this particular expression but also gain confidence in tackling similar mathematical problems. So, let’s dive in and simplify this expression together!

Understanding the Order of Operations

Before we delve into the specifics of our expression, it's essential to understand the order of operations. This is a set of rules that dictate the sequence in which mathematical operations should be performed. The most commonly used mnemonic for this order is PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) or BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction). Both acronyms represent the same hierarchy of operations. Let's break it down:

  1. Parentheses/Brackets (P/B): Operations inside parentheses or brackets are always performed first. This is because parentheses group terms together, and whatever is inside must be simplified as a single unit before interacting with anything outside. This is a fundamental rule, ensuring that expressions within are treated as a priority, allowing for a clear and logical progression in solving the overall mathematical statement.

  2. Exponents/Orders (E/O): Next, we handle exponents or orders (powers and square roots). These operations raise a number to a certain power or find the root of a number. Exponents signify repeated multiplication, and their correct evaluation is crucial. For example, 3^2 (3 squared) means 3 multiplied by itself, which equals 9. Understanding how to evaluate exponents is a vital step, as it often simplifies the subsequent calculations and ensures the accuracy of the final result.

  3. Multiplication and Division (MD): Multiplication and division are performed from left to right. These operations have equal precedence, so we perform them in the order they appear. This left-to-right rule is essential when both operations are present to avoid ambiguity. Multiplication and division are inverse operations, and their proper sequencing ensures that the expression is simplified logically and accurately, leading to the correct outcome.

  4. Addition and Subtraction (AS): Finally, addition and subtraction are performed from left to right, similar to multiplication and division. These are also inverse operations and are done in the order they appear from left to right. Ensuring that these operations are executed in the correct order maintains the mathematical integrity of the expression and provides a clear path to the final simplified answer.

Following this order ensures that we evaluate mathematical expressions consistently and accurately. Neglecting the correct order can lead to incorrect results, so it is crucial to adhere to these rules. In our expression extbf{(7+1)2-(11+32) ÷ 4}, we will methodically apply PEMDAS/BODMAS to arrive at the correct solution.

Step-by-Step Simplification

Now that we understand the order of operations, let's apply it to our expression: extbf{(7+1)2-(11+32) ÷ 4}. We'll go through each step methodically to ensure clarity.

Step 1: Parentheses

First, we deal with the expressions inside the parentheses. We have two sets of parentheses: (7+1) and (11+3^2). Let's simplify them one at a time.

  • (7+1): This is a simple addition. 7 + 1 = 8. So, the first set of parentheses simplifies to 8.
  • (11+3^2): Inside this set, we have both addition and an exponent. According to PEMDAS/BODMAS, we must handle the exponent first. 3^2 (3 squared) is 3 * 3 = 9. Now we have (11 + 9), which simplifies to 20.

After simplifying the expressions within the parentheses, our expression now looks like this:

8^2 - 20 ÷ 4

Step 2: Exponents

The next step is to address any exponents. In our simplified expression, we have 8^2. This means 8 raised to the power of 2, which is 8 * 8 = 64. So, 8^2 simplifies to 64.

Our expression now becomes:

64 - 20 ÷ 4

Step 3: Division

Now, we perform any multiplication and division, working from left to right. In our expression, we only have one division operation: 20 ÷ 4. Dividing 20 by 4 gives us 5.

So, our expression now looks like this:

64 - 5

Step 4: Subtraction

Finally, we perform the remaining subtraction operation. 64 - 5 equals 59.

Therefore, the simplified form of the expression extbf{(7+1)2-(11+32) ÷ 4} is 59.

Detailed Breakdown of Calculations

To further clarify the simplification process, let's break down each calculation step by step. This detailed breakdown will reinforce your understanding and provide a clear reference for similar problems.

  1. Simplify (7+1):

    • 7 + 1 = 8

  2. Simplify (11+3^2):

    • First, evaluate the exponent: 3^2 = 3 * 3 = 9
    • Then, add: 11 + 9 = 20

  3. Evaluate 8^2:

    • 8^2 = 8 * 8 = 64

  4. Perform the division 20 ÷ 4:

    • 20 ÷ 4 = 5

  5. Perform the subtraction 64 - 5:

    • 64 - 5 = 59

By breaking down each step, we can see exactly how the order of operations is applied. This methodical approach not only helps in solving the problem accurately but also builds a strong foundation for tackling more complex mathematical expressions. The key is to handle each operation in its correct order, following the PEMDAS/BODMAS rules.

Common Mistakes to Avoid

When simplifying mathematical expressions, it's easy to make mistakes if the order of operations is not followed correctly. Identifying these common pitfalls can help you avoid errors and improve your accuracy. Let's discuss some typical mistakes to watch out for:

  1. Ignoring the Order of Operations: One of the most common mistakes is not following the correct order (PEMDAS/BODMAS). For example, adding before multiplying or dividing before dealing with exponents. In our expression, someone might mistakenly subtract 20 ÷ 4 from 64 before performing the division, leading to an incorrect result.

  2. Incorrectly Handling Parentheses: Failing to simplify expressions within parentheses first can also lead to errors. It’s crucial to treat the content within parentheses as a single unit before performing any operations outside. In our problem, if the exponent in (11+3^2) is not calculated before adding 11, it will result in a wrong answer.

  3. Misunderstanding Exponents: Exponents represent repeated multiplication, not simple multiplication by the base. For instance, 3^2 means 3 multiplied by itself (3 * 3), not 3 multiplied by 2. Misinterpreting exponents can lead to significant calculation errors.

  4. Forgetting to Work Left to Right: For operations with equal precedence (like multiplication and division, or addition and subtraction), it's essential to work from left to right. This ensures consistent and correct evaluation. For example, if an expression had both multiplication and division, performing the rightmost operation first, instead of the leftmost, can lead to a wrong result.

  5. Calculation Errors: Simple arithmetic errors can easily occur, especially in multi-step problems. Double-checking each calculation can help catch these mistakes. This includes ensuring that addition, subtraction, multiplication, and division are performed accurately at each step.

By being aware of these common mistakes, you can take extra care to avoid them. Always remember to follow the order of operations, handle parentheses correctly, understand exponents, work from left to right for operations of equal precedence, and double-check your calculations. Avoiding these pitfalls will significantly improve your accuracy in simplifying mathematical expressions.

Practice Problems

To solidify your understanding of simplifying expressions using the order of operations, let's work through some practice problems. These exercises will help you apply the concepts we've discussed and build your confidence in tackling mathematical expressions.

Practice Problem 1:

Simplify: extbf{10 + 2 * (15 - 5) ÷ 4}

  1. First, simplify the expression inside the parentheses: (15 - 5) = 10
  2. The expression becomes: 10 + 2 * 10 ÷ 4
  3. Next, perform multiplication: 2 * 10 = 20
  4. The expression becomes: 10 + 20 ÷ 4
  5. Now, perform division: 20 ÷ 4 = 5
  6. The expression becomes: 10 + 5
  7. Finally, perform addition: 10 + 5 = 15

So, the simplified form of 10 + 2 * (15 - 5) ÷ 4 is 15.

Practice Problem 2:

Simplify: extbf{(4 + 2)^2 - 16 ÷ 2 + 3}

  1. First, simplify the expression inside the parentheses: (4 + 2) = 6
  2. The expression becomes: 6^2 - 16 ÷ 2 + 3
  3. Next, handle the exponent: 6^2 = 36
  4. The expression becomes: 36 - 16 ÷ 2 + 3
  5. Now, perform division: 16 ÷ 2 = 8
  6. The expression becomes: 36 - 8 + 3
  7. Perform subtraction: 36 - 8 = 28
  8. The expression becomes: 28 + 3
  9. Finally, perform addition: 28 + 3 = 31

So, the simplified form of (4 + 2)^2 - 16 ÷ 2 + 3 is 31.

Practice Problem 3:

Simplify: extbf{5 * (12 - 2^2) + 18 ÷ 3}

  1. First, simplify the expression inside the parentheses. Within the parentheses, handle the exponent first: 2^2 = 4
  2. The expression within the parentheses becomes: (12 - 4) = 8
  3. The expression becomes: 5 * 8 + 18 ÷ 3
  4. Next, perform multiplication: 5 * 8 = 40
  5. The expression becomes: 40 + 18 ÷ 3
  6. Now, perform division: 18 ÷ 3 = 6
  7. The expression becomes: 40 + 6
  8. Finally, perform addition: 40 + 6 = 46

So, the simplified form of 5 * (12 - 2^2) + 18 ÷ 3 is 46.

These practice problems demonstrate the application of the order of operations in different scenarios. Remember to always follow PEMDAS/BODMAS to ensure accurate simplification. Consistent practice will help you become more proficient in solving these types of mathematical expressions.

Conclusion

In conclusion, simplifying the expression extbf{(7+1)2-(11+32) ÷ 4} involves a systematic approach using the order of operations (PEMDAS/BODMAS). We first addressed the expressions within parentheses, then handled the exponents, followed by division, and finally subtraction. By meticulously following each step, we arrived at the simplified answer of 59.

Throughout this article, we emphasized the importance of understanding and correctly applying the order of operations to avoid common mistakes. We broke down each step of the simplification process, provided a detailed breakdown of calculations, and discussed common pitfalls to watch out for. Additionally, we worked through several practice problems to reinforce these concepts and build your problem-solving skills.

Mastering the order of operations is crucial for success in mathematics. It's not just about getting the right answer; it's about understanding the process and developing a logical approach to problem-solving. With consistent practice and attention to detail, you can confidently simplify complex mathematical expressions and enhance your overall mathematical proficiency. Remember, the key is to follow PEMDAS/BODMAS, work methodically, and double-check your calculations to ensure accuracy.