Solving 3-2[4-2]+(8)+4(2)+5 A Step-by-Step Guide

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In this comprehensive guide, we will walk you through the process of solving the mathematical expression 3-2[4-2]+(8)+4(2)+5 step by step. This expression involves various arithmetic operations, including subtraction, multiplication, and addition, as well as the use of brackets and parentheses. By following the order of operations (PEMDAS/BODMAS), we will break down the problem into manageable steps and arrive at the correct solution. Whether you're a student learning basic arithmetic or someone looking to refresh your math skills, this guide will provide a clear and easy-to-understand explanation of how to solve this type of expression.

Understanding the Order of Operations (PEMDAS/BODMAS)

Before we dive into solving the expression 3-2[4-2]+(8)+4(2)+5, it's crucial to understand the order of operations. This is a set of rules that dictate the sequence in which mathematical operations should be performed. The acronyms PEMDAS and BODMAS are commonly used to remember this order.

  • PEMDAS stands for: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).
  • BODMAS stands for: Brackets, Orders, Division and Multiplication (from left to right), Addition and Subtraction (from left to right).

Both acronyms represent the same order of operations. The key takeaway is that we must address operations within parentheses or brackets first, followed by exponents or orders, then multiplication and division (from left to right), and finally addition and subtraction (from left to right). Understanding and applying this order is essential for accurate mathematical calculations.

Step 1: Simplify within Brackets and Parentheses

The given expression is 3-2[4-2]+(8)+4(2)+5. According to the order of operations, our first step is to simplify any expressions within brackets and parentheses. In this case, we have one set of brackets, [4-2], and one set of parentheses, (8) and (2). Let's simplify them:

  • [4-2]: Subtract 2 from 4, which equals 2. So, [4-2] = 2.
  • (8): The parentheses around 8 don't change its value, so (8) = 8.
  • (2): Similarly, the parentheses around 2 don't change its value, so (2) = 2.

Now, substitute these simplified values back into the original expression. The expression becomes 3-2[2]+8+4(2)+5. We have successfully simplified the expressions within the brackets and parentheses, paving the way for the next steps in solving the equation.

Step 2: Perform Multiplication

After simplifying the expressions within brackets and parentheses, the next step in the order of operations is to perform multiplication. Our expression now looks like this: 3-2[2]+8+4(2)+5. We have two multiplication operations to address:

  • -2[2]: This means -2 multiplied by 2, which equals -4. So, -2[2] = -4.
  • 4(2): This means 4 multiplied by 2, which equals 8. So, 4(2) = 8.

Now, substitute these results back into the expression. The expression becomes 3-4+8+8+5. We have successfully performed the multiplication operations, further simplifying the expression and bringing us closer to the final solution. Remember to pay close attention to signs (positive and negative) during multiplication to avoid errors.

Step 3: Perform Addition and Subtraction (from left to right)

With the multiplication completed, we are now left with addition and subtraction operations. According to the order of operations, we perform these operations from left to right. Our expression is now 3-4+8+8+5.

Let's break it down step by step:

  1. 3-4: Subtract 4 from 3, which equals -1. So, the expression becomes -1+8+8+5.
  2. -1+8: Add 8 to -1, which equals 7. So, the expression becomes 7+8+5.
  3. 7+8: Add 8 to 7, which equals 15. So, the expression becomes 15+5.
  4. 15+5: Finally, add 5 to 15, which equals 20.

Therefore, the final result of the expression 3-4+8+8+5 is 20. We have successfully performed all the addition and subtraction operations from left to right, adhering to the correct order of operations.

Final Solution

By following the order of operations (PEMDAS/BODMAS) and breaking down the expression 3-2[4-2]+(8)+4(2)+5 into manageable steps, we have arrived at the final solution.

  • Step 1: Simplify within brackets and parentheses: 3-2[2]+8+4(2)+5
  • Step 2: Perform multiplication: 3-4+8+8+5
  • Step 3: Perform addition and subtraction (from left to right): 20

Therefore, the solution to the expression 3-2[4-2]+(8)+4(2)+5 is 20. This step-by-step guide demonstrates the importance of following the correct order of operations to ensure accurate mathematical calculations. Understanding and applying PEMDAS/BODMAS is fundamental to solving more complex mathematical problems.

Common Mistakes to Avoid

When solving mathematical expressions, it's easy to make mistakes if you're not careful. Here are some common mistakes to avoid when working with expressions like 3-2[4-2]+(8)+4(2)+5:

  1. Ignoring the Order of Operations: This is the most common mistake. Failing to follow PEMDAS/BODMAS can lead to incorrect answers. For example, adding before multiplying or subtracting before dividing will result in a wrong solution. Always remember to address parentheses/brackets, exponents/orders, multiplication and division (from left to right), and addition and subtraction (from left to right).

  2. Incorrectly Handling Negative Signs: Pay close attention to negative signs, especially during multiplication and subtraction. For instance, -2[2] should be calculated as -4, not 4. Similarly, when subtracting a negative number, remember that it's the same as adding a positive number.

  3. Misinterpreting Parentheses and Brackets: Parentheses and brackets indicate the order in which operations should be performed. Make sure to simplify the expressions within them before proceeding with other operations. Forgetting to do this can alter the entire calculation.

  4. Arithmetic Errors: Simple arithmetic mistakes, such as incorrect addition, subtraction, multiplication, or division, can derail the entire solution. Double-check each calculation to minimize the risk of errors.

  5. Skipping Steps: Trying to solve the expression too quickly by skipping steps can lead to mistakes. Break down the problem into smaller, manageable steps to reduce the chances of errors. Each step should be written down clearly to keep track of the calculations.

By being aware of these common mistakes and taking the time to work through the expression carefully, you can improve your accuracy and confidence in solving mathematical problems. Remember, practice makes perfect, so keep working on similar expressions to hone your skills.