Solve Order Of Operations: 5² + 2[73 - (4 × 5)]

by Luna Greco 48 views

Hey everyone! Today, we're diving into the fascinating world of mathematics, specifically focusing on the order of operations. This is a fundamental concept that helps us solve complex mathematical expressions in the correct sequence. Imagine trying to bake a cake without following the recipe – you might end up with a delicious mess, but it probably won't be the cake you were hoping for. Similarly, in math, if we don't follow the order of operations, we might arrive at the wrong answer. So, let's unravel this mathematical recipe together and become masters of order of operations! Our specific challenge today is to solve the expression: 5² + 2[73 - (4 × 5)]. This looks a bit intimidating at first glance, but don't worry, we'll break it down step by step using a simple and effective mnemonic.

Understanding the Order of Operations: PEMDAS/BODMAS

So, how do we tackle this mathematical puzzle? The key lies in understanding the order of operations, which is often remembered by the acronyms PEMDAS or BODMAS. These acronyms serve as our guide, telling us the precise order in which to perform mathematical operations. Let's break down what each letter stands for:

  • P - Parentheses (or B - Brackets): This is our starting point. Any operations within parentheses or brackets must be performed first. Think of it as clearing the inner layers before moving to the outer ones. Parentheses and brackets act as containers, grouping operations together and dictating the priority. We need to simplify everything inside these containers before moving on.
  • E - Exponents (or O - Orders): Next up are exponents. These are the little numbers that tell us how many times to multiply a number by itself (e.g., 5² means 5 multiplied by itself). Exponents represent repeated multiplication and hold a higher priority than regular multiplication or division.
  • MD - Multiplication and Division: These two operations are on the same level of priority. We perform them from left to right, whichever comes first. It's like reading a sentence – we tackle the operations in the order we encounter them.
  • AS - Addition and Subtraction: Similarly, addition and subtraction have the same priority level. We perform them from left to right, just like reading.

Remembering this order is crucial for accurate calculations. PEMDAS/BODMAS provides a clear roadmap, ensuring we don't get lost in the maze of numbers and operations. Without it, we might be tempted to perform operations in the wrong sequence, leading to incorrect results. So, keep this mnemonic in mind as we move forward and apply it to our specific problem.

Step-by-Step Solution of 5² + 2[73 - (4 × 5)]

Now that we have our trusty PEMDAS/BODMAS guide, let's apply it to the expression 5² + 2[73 - (4 × 5)]. We'll break it down step by step, showing each operation and its result. This methodical approach will not only give us the correct answer but also solidify our understanding of the order of operations. Let's get started!

Step 1: Parentheses/Brackets

Our first task is to tackle the innermost parentheses: (4 × 5). This is a straightforward multiplication, so we simply multiply 4 by 5, which gives us 20. The expression now looks like this: 5² + 2[73 - 20].

Next, we have the brackets [73 - 20]. This is a subtraction operation. Subtracting 20 from 73 gives us 53. Our expression is now simplified to: 5² + 2[53]. Notice that the brackets are still there, indicating that we need to deal with the multiplication next.

Step 2: Exponents

Moving on to exponents, we have 5², which means 5 multiplied by itself (5 × 5). This equals 25. Our expression now becomes: 25 + 2[53]. We're making good progress! The expression is becoming less complex, and we're one step closer to the final answer.

Step 3: Multiplication and Division

Now we encounter multiplication. We have 2[53], which means 2 multiplied by 53. This gives us 106. Our expression is further simplified to: 25 + 106. We've successfully handled the multiplication, and we're left with a simple addition problem.

Step 4: Addition and Subtraction

Finally, we have addition. We need to add 25 and 106. This gives us the final result: 131. So, the solution to the expression 5² + 2[73 - (4 × 5)] is 131.

By following PEMDAS/BODMAS meticulously, we've successfully navigated through the expression and arrived at the correct answer. Each step was crucial, and by understanding the order of operations, we avoided any potential pitfalls. This methodical approach can be applied to any mathematical expression, no matter how complex it may seem.

Common Mistakes to Avoid

Even with a clear understanding of PEMDAS/BODMAS, it's easy to make mistakes if we're not careful. Let's discuss some common pitfalls to avoid, ensuring we become even more proficient at solving mathematical expressions. Recognizing these errors will help us develop a more robust and accurate approach to problem-solving.

Mistake 1: Ignoring the Order

The most common mistake is simply not following the correct order of operations. For example, someone might be tempted to add 5² and 2 first in our problem, but that would lead to an incorrect result. Remember, exponents and multiplication/division take precedence over addition and subtraction. Always double-check that you're following the PEMDAS/BODMAS sequence.

Mistake 2: Misinterpreting Parentheses/Brackets

Another frequent error is not fully simplifying the expressions within parentheses or brackets before moving on. Make sure you've performed all operations within the parentheses/brackets, even if they involve multiple steps themselves. The parentheses act as a container, and we need to empty the container completely before proceeding.

Mistake 3: Incorrectly Handling Multiplication and Division or Addition and Subtraction

Remember, multiplication and division have equal priority, as do addition and subtraction. This means we perform these operations from left to right. A common mistake is to always perform multiplication before division or addition before subtraction, which is not correct. The order in which they appear in the expression matters.

Mistake 4: Calculation Errors

Simple arithmetic errors can also derail our efforts. A wrong multiplication or addition can throw off the entire calculation. It's always a good idea to double-check your calculations, especially in longer expressions. Consider using a calculator for complex calculations to minimize the risk of errors.

Mistake 5: Not Showing Your Work

This might seem like a minor point, but not showing your work can make it difficult to spot mistakes. When you write down each step, it's easier to review your process and identify any errors. Showing your work also helps you understand the problem better and reinforces your understanding of the order of operations.

By being aware of these common mistakes and taking steps to avoid them, we can significantly improve our accuracy and confidence in solving mathematical expressions. Practice is key, so keep applying PEMDAS/BODMAS to different problems, and you'll become a master of order of operations in no time!

Practice Makes Perfect: More Examples

To truly master the order of operations, it's essential to practice with a variety of examples. Let's work through a few more expressions together, reinforcing our understanding of PEMDAS/BODMAS. Each example will present a slightly different challenge, helping us refine our skills and build confidence. Remember, the more we practice, the more natural the order of operations will become.

Example 1: 10 + 3 × (12 - 4) ÷ 2

  1. Parentheses: First, we tackle the parentheses: (12 - 4) = 8. The expression becomes: 10 + 3 × 8 ÷ 2.
  2. Multiplication and Division (from left to right): Next, we have multiplication and division. We perform them from left to right. First, 3 × 8 = 24. The expression becomes: 10 + 24 ÷ 2. Then, 24 ÷ 2 = 12. The expression becomes: 10 + 12.
  3. Addition: Finally, we have addition: 10 + 12 = 22. So, the solution is 22.

Example 2: (6² + 4) ÷ 5 - 1

  1. Parentheses: Inside the parentheses, we first deal with the exponent: 6² = 36. The expression inside the parentheses becomes: (36 + 4). Then, 36 + 4 = 40. The expression becomes: 40 ÷ 5 - 1.
  2. Division: Next, we have division: 40 ÷ 5 = 8. The expression becomes: 8 - 1.
  3. Subtraction: Finally, we have subtraction: 8 - 1 = 7. So, the solution is 7.

Example 3: 2 × [15 - (3 + 2) × 2] + 8

  1. Innermost Parentheses: We start with the innermost parentheses: (3 + 2) = 5. The expression becomes: 2 × [15 - 5 × 2] + 8.
  2. Brackets: Inside the brackets, we have multiplication: 5 × 2 = 10. The expression inside the brackets becomes: [15 - 10]. Then, 15 - 10 = 5. The expression becomes: 2 × 5 + 8.
  3. Multiplication: Next, we have multiplication: 2 × 5 = 10. The expression becomes: 10 + 8.
  4. Addition: Finally, we have addition: 10 + 8 = 18. So, the solution is 18.

By working through these examples, we've seen how PEMDAS/BODMAS guides us through different types of expressions. Each example highlights the importance of following the order meticulously. Keep practicing with more examples, and you'll become a confident problem-solver in mathematics!

Conclusion: The Power of Order

In conclusion, mastering the order of operations is a fundamental skill in mathematics. By understanding and applying PEMDAS/BODMAS, we can confidently solve complex expressions and avoid common pitfalls. We've seen how breaking down problems step by step, following the correct sequence, leads to accurate results. Whether it's a simple arithmetic problem or a more intricate algebraic equation, the order of operations provides a clear roadmap for success.

Remember, mathematics is like a language, and the order of operations is its grammar. Just as we follow grammatical rules to construct meaningful sentences, we follow PEMDAS/BODMAS to construct accurate mathematical solutions. The more we practice, the more fluent we become in this language, and the more confident we feel tackling mathematical challenges.

So, embrace the power of order, keep practicing, and continue exploring the fascinating world of mathematics. With a solid understanding of the order of operations, you'll be well-equipped to tackle any mathematical problem that comes your way. And who knows, you might even find a newfound appreciation for the beauty and logic of numbers! Keep up the great work, guys, and happy problem-solving!