Simplifying Complex Expressions A Step By Step Guide

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In the realm of mathematics, simplifying complex expressions is a fundamental skill. It involves applying the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), to arrive at a single, simplified value. Mastering this skill is crucial for success in various mathematical disciplines, from basic arithmetic to advanced calculus. This article will walk you through a step-by-step simplification of the expression 45 + 3{34 - 18 - 14} ÷ 3 [17 + 3 × 4 - (2 × 1)] × 9 ÷ 5 × 6, highlighting the key principles and techniques involved.

Understanding the Order of Operations (PEMDAS/BODMAS)

Before diving into the simplification process, it's essential to understand the order of operations. This set of rules dictates the sequence in which mathematical operations should be performed to ensure a consistent and accurate result. The acronyms PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) and BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction) are commonly used to remember this order.

  • Parentheses/Brackets: Operations within parentheses or brackets are performed first. This includes all types of brackets: parentheses (), square brackets [], and curly braces {}.
  • Exponents/Orders: Next, exponents or orders (powers and square roots, etc.) are evaluated.
  • Multiplication and Division: Multiplication and division are performed from left to right.
  • Addition and Subtraction: Finally, addition and subtraction are performed from left to right.

Adhering to this order is crucial for obtaining the correct answer. Let's now apply these principles to simplify the given expression.

Step-by-Step Simplification

To simplify the expression 45 + 3{34 - 18 - 14} ÷ 3 [17 + 3 × 4 - (2 × 1)] × 9 ÷ 5 × 6, we'll follow the order of operations meticulously.

1. Simplify within Parentheses and Braces

We begin by simplifying the expressions within the parentheses and braces, starting with the innermost ones:

  • (2 × 1) = 2
  • {34 - 18 - 14} = {16 - 14} = 2
  • [17 + 3 × 4 - (2 × 1)] = [17 + 3 × 4 - 2]

Now, the expression looks like this: 45 + 3{2} ÷ 3 [17 + 3 × 4 - 2] × 9 ÷ 5 × 6

2. Perform Multiplication within Brackets

Next, we perform the multiplication within the square brackets:

  • 3 × 4 = 12
  • [17 + 12 - 2]

Now, the expression inside the square brackets simplifies to:

  • [17 + 12 - 2] = [29 - 2] = 27

Our expression now becomes: 45 + 3{2} ÷ 3 [27] × 9 ÷ 5 × 6

3. Perform Multiplication and Division (Left to Right)

Now, we perform the multiplication and division operations from left to right:

  • 3{2} = 3 × 2 = 6
  • 6 ÷ 3 = 2
  • 2 [27] = 2 × 27 = 54
  • 54 × 9 = 486
  • 486 ÷ 5 = 97.2
  • 97.2 × 6 = 583.2

Our expression is now reduced to: 45 + 583.2

4. Perform Addition

Finally, we perform the addition:

  • 45 + 583.2 = 628.2

Therefore, the simplified value of the expression 45 + 3{34 - 18 - 14} ÷ 3 [17 + 3 × 4 - (2 × 1)] × 9 ÷ 5 × 6 is 628.2.

Detailed Breakdown of Each Step

To further clarify the simplification process, let's break down each step with additional explanations:

Step 1: Simplifying Parentheses and Braces

This step is crucial as it addresses the innermost operations first, ensuring the correct order of evaluation. We start with the parentheses (2 × 1), which is a straightforward multiplication resulting in 2. Then, we move to the curly braces {34 - 18 - 14}. Inside the braces, we perform subtraction from left to right: 34 - 18 = 16, and then 16 - 14 = 2. This simplifies the expression within the curly braces to 2. Finally, we address the square brackets [17 + 3 × 4 - (2 × 1)]. We've already simplified (2 × 1) to 2, so the expression inside the brackets becomes [17 + 3 × 4 - 2]. This sets the stage for the next step, where we'll handle the multiplication within the brackets.

Step 2: Performing Multiplication within Brackets

Within the square brackets, we encounter the multiplication operation 3 × 4. According to the order of operations, multiplication takes precedence over addition and subtraction. Therefore, we multiply 3 by 4, which gives us 12. The expression inside the brackets now looks like this: [17 + 12 - 2]. This simplification is crucial as it reduces the complexity of the expression within the brackets, making it easier to handle in the subsequent steps. Next, we will perform the addition and subtraction operations from left to right.

Step 3: Performing Addition and Subtraction within Brackets

Continuing with the square brackets [17 + 12 - 2], we now perform addition and subtraction from left to right. First, we add 17 and 12, which results in 29. The expression then becomes [29 - 2]. Subtracting 2 from 29 gives us 27. Therefore, the expression within the square brackets simplifies to 27. This step is vital as it condenses the expression within the brackets to a single numerical value, which can then be used in the subsequent multiplication and division operations.

Step 4: Performing Multiplication and Division (Left to Right)

With the expressions within the parentheses, braces, and brackets simplified, we now focus on multiplication and division. The expression is 45 + 3{2} ÷ 3 [27] × 9 ÷ 5 × 6. We begin by evaluating 3{2}, which means 3 multiplied by 2, resulting in 6. The expression becomes 45 + 6 ÷ 3 [27] × 9 ÷ 5 × 6. Next, we perform the division 6 ÷ 3, which equals 2. The expression is now 45 + 2 [27] × 9 ÷ 5 × 6. We then multiply 2 by 27, which gives us 54. The expression becomes 45 + 54 × 9 ÷ 5 × 6. Continuing with multiplication and division from left to right, we multiply 54 by 9, resulting in 486. The expression becomes 45 + 486 ÷ 5 × 6. Dividing 486 by 5 gives us 97.2. The expression is now 45 + 97.2 × 6. Finally, we multiply 97.2 by 6, which equals 583.2. The expression is reduced to 45 + 583.2. This step involves multiple operations, and careful attention to the order of operations is crucial to arrive at the correct result.

Step 5: Performing Addition

The final step is to perform the addition. We have the expression 45 + 583.2. Adding these two numbers gives us 628.2. This is the simplified value of the original expression. This final step is straightforward but essential to complete the simplification process and arrive at the final answer.

Common Mistakes to Avoid

Simplifying complex expressions can be tricky, and it's easy to make mistakes if you're not careful. Here are some common mistakes to avoid:

  1. Ignoring the Order of Operations: This is the most common mistake. Always follow PEMDAS/BODMAS to ensure you perform operations in the correct order.
  2. Incorrectly Handling Negative Signs: Pay close attention to negative signs, especially when dealing with subtraction or multiplication of negative numbers.
  3. Skipping Steps: It's tempting to skip steps to save time, but this can lead to errors. Write out each step clearly to minimize mistakes.
  4. Misinterpreting Parentheses and Brackets: Make sure you understand the hierarchy of parentheses, square brackets, and curly braces. Simplify the innermost expressions first.
  5. Calculator Errors: While calculators can be helpful, they can also lead to errors if you don't input the expression correctly. Double-check your inputs and be mindful of the calculator's order of operations.

By being aware of these common mistakes, you can significantly improve your accuracy when simplifying expressions.

Practice Makes Perfect

Simplifying mathematical expressions is a skill that improves with practice. The more you practice, the more comfortable and confident you'll become. Here are some tips for effective practice:

  • Start with Simple Expressions: Begin with simpler expressions and gradually increase the complexity as you improve.
  • Work Through Examples: Study worked examples to understand the process and techniques involved.
  • Solve Practice Problems: Solve a variety of practice problems to reinforce your understanding.
  • Check Your Answers: Always check your answers to identify and correct any mistakes.
  • Seek Help When Needed: Don't hesitate to ask for help from teachers, tutors, or online resources if you're struggling.

Consistent practice is the key to mastering the art of simplifying mathematical expressions.

Conclusion

Simplifying the expression 45 + 3{34 - 18 - 14} ÷ 3 [17 + 3 × 4 - (2 × 1)] × 9 ÷ 5 × 6 demonstrates the importance of following the order of operations (PEMDAS/BODMAS) meticulously. By breaking down the expression into smaller, manageable steps and adhering to the correct sequence of operations, we arrived at the simplified value of 628.2. This process highlights the fundamental principles of mathematical simplification and the significance of accuracy and attention to detail. Mastering this skill is essential for success in mathematics and related fields. Remember, practice is key, and by consistently applying these principles, you can confidently tackle even the most complex expressions.