Evaluating Mathematical Expressions A Comprehensive Guide

This comprehensive guide provides a detailed, step-by-step explanation of how to evaluate mathematical expressions, ensuring clarity and understanding for learners of all levels. We will dissect each expression, applying the order of operations (PEMDAS/BODMAS) meticulously to arrive at the correct solution. Our focus will be on fostering a deep understanding of the underlying principles, empowering you to tackle complex calculations with confidence. This is a crucial skill in mathematics and numerous real-world applications. Understanding how to simplify expressions not only strengthens your mathematical foundation but also enhances your problem-solving abilities in various domains. So, let's embark on this journey of mathematical exploration, unraveling the intricacies of expression evaluation.

1. 2 × (24 - 4) ÷ 5 ÷ 2 [8 - 0 - (2 - 0 - 5)]

To evaluate this complex expression, we will meticulously follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) or BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction). This ensures we perform operations in the correct sequence, leading to the accurate result. Let's begin by focusing on the innermost parentheses and brackets, simplifying each step by step. Remember, accuracy is paramount, and each operation must be performed with precision to avoid errors. This systematic approach will break down the complexity into manageable steps.

Step 1: Simplify the Innermost Parentheses

Our initial focus is on simplifying the innermost parentheses: (2 - 0 - 5). This involves basic subtraction. Performing the subtraction from left to right, we get:

2 - 0 - 5 = 2 - 5 = -3

Now, let's replace the original parentheses with this simplified value in the expression:

2 × (24 - 4) ÷ 5 ÷ 2 [8 - 0 - (-3)]

This substitution is a crucial step, maintaining the integrity of the expression while reducing its complexity. The negative sign within the brackets now requires careful attention in the subsequent steps.

Step 2: Simplify the Brackets

Next, we simplify the expression within the brackets: [8 - 0 - (-3)]. Remember that subtracting a negative number is equivalent to adding its positive counterpart. Therefore, we have:

8 - 0 - (-3) = 8 + 3 = 11

Substituting this simplified value back into the expression, we get:

2 × (24 - 4) ÷ 5 ÷ 2 × 11

The brackets are now fully simplified, allowing us to move on to the next level of operations. The expression is becoming more manageable with each step.

Step 3: Simplify the Remaining Parentheses

Now, let's address the remaining parentheses: (24 - 4). This is a straightforward subtraction:

24 - 4 = 20

Replace the parentheses with the result:

2 × 20 ÷ 5 ÷ 2 × 11

The expression is further simplified, and we are now left with only multiplication and division operations.

Step 4: Perform Multiplication and Division (from left to right)

According to the order of operations, we perform multiplication and division from left to right. First, we multiply 2 by 20:

2 × 20 = 40

The expression becomes:

40 ÷ 5 ÷ 2 × 11

Next, divide 40 by 5:

40 ÷ 5 = 8

The expression is now:

8 ÷ 2 × 11

Now, divide 8 by 2:

8 ÷ 2 = 4

The expression simplifies to:

4 × 11

Step 5: Final Multiplication

Finally, perform the last multiplication:

4 × 11 = 44

Therefore, the value of the expression 2 × (24 - 4) ÷ 5 ÷ 2 [8 - 0 - (2 - 0 - 5)] is 44. This completes the evaluation of the first expression. Remember the importance of each step and the meticulous application of the order of operations.

2. 6 + 2 + 5

Evaluating this simple expression involving only addition is relatively straightforward. Addition is an associative operation, meaning the order in which we add the numbers does not affect the final result. However, for clarity and consistency, we will perform the addition from left to right. This is a fundamental arithmetic operation, and a clear understanding of addition is essential for more complex calculations. This example serves as a reminder of the basic building blocks of mathematical operations.

Step 1: Add the First Two Numbers

First, we add the first two numbers, 6 and 2:

6 + 2 = 8

Now, we replace the first two numbers with their sum in the expression:

8 + 5

Step 2: Add the Result to the Last Number

Next, we add the result from the previous step, 8, to the last number, 5:

8 + 5 = 13

Therefore, the value of the expression 6 + 2 + 5 is 13. This straightforward example reinforces the basic principles of addition. The simplicity of this example highlights the importance of mastering fundamental operations before tackling more complex expressions.

3. 4 × (6 + 8) - 2] - 2 ÷ 4 (5 + 7) - 2

This expression requires a careful application of the order of operations (PEMDAS/BODMAS) due to the presence of parentheses, multiplication, subtraction, and division. To evaluate this expression correctly, we will proceed step-by-step, simplifying the innermost parts first. The complexity of this expression underscores the importance of following the correct order of operations. Neglecting this order can lead to incorrect results. Let's break down the expression and tackle each part systematically.

Step 1: Simplify the Parentheses

We begin by simplifying the expressions within the parentheses. There are two sets of parentheses: (6 + 8) and (5 + 7). Let's simplify each one individually:

6 + 8 = 14

5 + 7 = 12

Now, substitute these results back into the expression:

4 × 14 - 2] - 2 ÷ 4 × 12 - 2

This substitution simplifies the expression and allows us to proceed with the next level of operations.

Step 2: Perform Multiplication

Next, we perform the multiplication operations. There are two multiplications in the expression: 4 × 14 and 2 ÷ 4 × 12. Remember to perform multiplication and division from left to right. First, we multiply 4 by 14:

4 × 14 = 56

The expression now becomes:

56 - 2] - 2 ÷ 4 × 12 - 2

Now, we address the division and multiplication part: 2 ÷ 4 × 12. Following the left-to-right rule, we first divide 2 by 4:

2 ÷ 4 = 0.5

Then, we multiply the result by 12:

0. 5 × 12 = 6

Substituting this back into the expression, we get:

56 - 2] - 6 - 2

Step 3: Perform Subtraction (from left to right)

Now, we perform the subtraction operations from left to right. First, subtract 2 from 56:

56 - 2 = 54

The expression simplifies to:

54 - 6 - 2

Next, subtract 6 from 54:

54 - 6 = 48

The expression becomes:

48 - 2

Step 4: Final Subtraction

Finally, perform the last subtraction:

48 - 2 = 46

Therefore, the value of the expression 4 × (6 + 8) - 2] - 2 ÷ 4 (5 + 7) - 2 is 46. This completes the evaluation of the third expression. The complexity of this expression highlights the importance of meticulous attention to detail and the consistent application of the order of operations.

Conclusion

In conclusion, this guide has provided a comprehensive, step-by-step approach to evaluating mathematical expressions. We have meticulously dissected each expression, applying the order of operations (PEMDAS/BODMAS) to arrive at the correct solutions. The importance of following the correct order of operations cannot be overstated, as it is the key to accurate calculations. This skill is fundamental not only in mathematics but also in various fields that require logical and analytical thinking. By mastering the principles outlined in this guide, you will be well-equipped to tackle a wide range of mathematical problems with confidence and precision. Remember, practice is essential to solidify your understanding and enhance your problem-solving abilities. Embrace the challenges, and continue to explore the fascinating world of mathematics.