Evaluate The Expression (64 ÷ 8 + 2 - 10) + (9 × 9 + 19) A Step-by-Step Guide

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In the realm of mathematics, evaluating 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 the correct answer. This article delves into the step-by-step process of evaluating the expression (64 ÷ 8 + 2 - 10) + (9 × 9 + 19), providing a clear and concise explanation for each operation. Whether you're a student learning the basics or someone looking to refresh your math skills, this guide will equip you with the knowledge to tackle similar problems with confidence. Let's embark on this mathematical journey and unravel the solution together.

Understanding the Order of Operations (PEMDAS/BODMAS)

Before we dive into solving the expression, it's crucial to understand the order of operations, often remembered by the acronyms PEMDAS or BODMAS. These acronyms serve as a roadmap, guiding us through the sequence in which mathematical operations should be performed to ensure we arrive at the correct answer. PEMDAS stands for Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). BODMAS, a similar acronym, represents Brackets, Orders (exponents and roots), Division and Multiplication (from left to right), Addition and Subtraction (from left to right). While the acronyms differ slightly, the underlying principle remains the same: perform 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).

The importance of following the order of operations cannot be overstated. Without a consistent approach, the same expression can yield different results, leading to confusion and errors. Imagine, for instance, if we were to perform addition before multiplication. The outcome would be drastically different, highlighting the need for a standardized method. The order of operations provides this standardization, ensuring that mathematical expressions are evaluated consistently and accurately, regardless of who is performing the calculation. This consistency is paramount in various fields, from basic arithmetic to advanced scientific and engineering applications.

Understanding and applying the order of operations is a cornerstone of mathematical proficiency. It forms the basis for more complex mathematical concepts and problem-solving. By adhering to PEMDAS/BODMAS, we not only ensure the accuracy of our calculations but also develop a systematic and logical approach to problem-solving, a skill that extends far beyond the realm of mathematics. This foundation allows us to approach mathematical challenges with confidence and precision, paving the way for further exploration and mastery of mathematical principles. Remember, mastering the order of operations is not just about memorizing an acronym; it's about understanding the underlying logic and applying it consistently to achieve accurate results.

Step-by-Step Solution of (64 ÷ 8 + 2 - 10) + (9 × 9 + 19)

Now, let's apply the order of operations to solve the given expression: (64 ÷ 8 + 2 - 10) + (9 × 9 + 19). We will break down the solution into manageable steps, ensuring clarity and precision at each stage. Our journey begins with simplifying the expressions within the parentheses, adhering to the PEMDAS/BODMAS guidelines.

Step 1: Simplify the first set of parentheses (64 ÷ 8 + 2 - 10)

Within this set of parentheses, we encounter division, addition, and subtraction. According to the order of operations, division takes precedence over addition and subtraction. Therefore, we begin by dividing 64 by 8:

64 ÷ 8 = 8

Now, our expression within the first set of parentheses becomes:

8 + 2 - 10

Next, we perform addition and subtraction from left to right. First, we add 8 and 2:

8 + 2 = 10

Our expression now simplifies to:

10 - 10

Finally, we subtract 10 from 10:

10 - 10 = 0

Thus, the first set of parentheses simplifies to 0.

Step 2: Simplify the second set of parentheses (9 × 9 + 19)

In the second set of parentheses, we have multiplication and addition. Following the order of operations, multiplication comes before addition. We begin by multiplying 9 by 9:

9 × 9 = 81

Our expression within the second set of parentheses now becomes:

81 + 19

Next, we add 81 and 19:

81 + 19 = 100

Therefore, the second set of parentheses simplifies to 100.

Step 3: Combine the results

Now that we have simplified both sets of parentheses, our original expression reduces to:

0 + 100

Finally, we add 0 and 100:

0 + 100 = 100

Thus, the final answer to the expression (64 ÷ 8 + 2 - 10) + (9 × 9 + 19) is 100.

Detailed Breakdown of Each Operation

To further solidify our understanding, let's dissect each operation performed in the evaluation of the expression (64 ÷ 8 + 2 - 10) + (9 × 9 + 19). This detailed breakdown will illuminate the application of the order of operations and reinforce the step-by-step approach to solving mathematical expressions.

1. Division (64 ÷ 8)

The first operation we encountered was division within the first set of parentheses: 64 ÷ 8. Division, one of the four basic arithmetic operations, involves splitting a quantity into equal parts. In this case, we are dividing 64 by 8, which means we are determining how many groups of 8 can be formed from 64. The result of this division is 8, as 8 multiplied by 8 equals 64. This operation sets the stage for the subsequent addition and subtraction within the parentheses.

2. Multiplication (9 × 9)

Moving to the second set of parentheses, we encountered multiplication: 9 × 9. Multiplication is a fundamental arithmetic operation that represents repeated addition. In this instance, we are multiplying 9 by itself, which is equivalent to adding 9 to itself 9 times. The product of 9 and 9 is 81. This result is crucial for the next operation, which is addition.

3. Addition (8 + 2)

Following the division within the first set of parentheses, we performed addition: 8 + 2. Addition is the process of combining two or more quantities to find their total. In this case, we are adding 8 and 2, resulting in a sum of 10. This operation is a stepping stone towards simplifying the expression further.

4. Subtraction (10 - 10)

Next, we encountered subtraction within the first set of parentheses: 10 - 10. Subtraction is the inverse operation of addition, representing the process of finding the difference between two quantities. Subtracting 10 from 10 yields a result of 0. This operation completes the simplification of the first set of parentheses.

5. Addition (81 + 19)

In the second set of parentheses, we performed addition after the multiplication: 81 + 19. As mentioned earlier, addition combines quantities to find their total. Adding 81 and 19 gives us a sum of 100. This operation simplifies the second set of parentheses completely.

6. Final Addition (0 + 100)

Finally, after simplifying both sets of parentheses, we combined the results using addition: 0 + 100. Adding 0 to any number does not change its value, so the sum of 0 and 100 is simply 100. This final operation provides us with the solution to the entire expression.

By meticulously examining each operation, we gain a deeper appreciation for the importance of the order of operations and the step-by-step approach to evaluating mathematical expressions. This detailed breakdown serves as a valuable tool for reinforcing our understanding and building confidence in our mathematical abilities.

Common Mistakes to Avoid

When evaluating mathematical expressions, it's easy to fall prey to common errors if we're not careful. Understanding these pitfalls can help us avoid them and ensure accurate calculations. Let's explore some frequent mistakes and how to steer clear of them.

1. Ignoring the Order of Operations: This is perhaps the most pervasive mistake. Failing to adhere to the order of operations (PEMDAS/BODMAS) can lead to drastically incorrect results. For instance, performing addition before multiplication or subtraction before division will yield the wrong answer. Always prioritize parentheses, exponents, multiplication and division (from left to right), and finally addition and subtraction (from left to right).

2. Incorrectly Distributing Negative Signs: Negative signs can be tricky, especially when dealing with parentheses. A common mistake is failing to distribute a negative sign properly across all terms within parentheses. For example, if an expression involves subtracting a group of terms, remember to change the sign of each term inside the parentheses. Double-check the distribution of negative signs to ensure accuracy.

3. Misinterpreting Exponents: Exponents indicate the number of times a base is multiplied by itself. A frequent error is multiplying the base by the exponent instead of raising the base to the power of the exponent. For example, 2³ means 2 × 2 × 2, not 2 × 3. Remember that exponents represent repeated multiplication, not a simple multiplication of the base and exponent.

4. Errors in Basic Arithmetic: Simple arithmetic mistakes, such as incorrect addition, subtraction, multiplication, or division, can derail the entire calculation. These errors often stem from carelessness or rushing through the steps. Take your time, double-check your arithmetic, and use estimation to verify the reasonableness of your answers.

5. Forgetting to Simplify Completely: After performing the necessary operations, it's crucial to simplify the expression as much as possible. This may involve combining like terms, reducing fractions, or performing further calculations. Failing to simplify completely can leave the answer in an unrefined form. Always strive to simplify the expression to its simplest form.

6. Misunderstanding Fraction Operations: Operations with fractions, such as addition, subtraction, multiplication, and division, require specific rules. A common mistake is adding or subtracting fractions without finding a common denominator. Ensure you understand the rules for fraction operations and apply them correctly.

By being aware of these common pitfalls and taking proactive steps to avoid them, we can significantly improve our accuracy in evaluating mathematical expressions. Remember, practice makes perfect, and consistent attention to detail is key to success.

Practice Problems for Reinforcement

To truly master the art of evaluating expressions, practice is essential. Working through various problems helps solidify your understanding of the order of operations and builds confidence in your problem-solving abilities. Here are some practice problems that will challenge your skills and reinforce the concepts we've discussed:

  1. (12 + 8) ÷ 4 - 2 × 3
  2. 15 - 3 × (4 + 1) + 7
  3. 24 ÷ (6 - 2) + 5² - 10
  4. (18 ÷ 3 + 2) × (7 - 4)
  5. 36 ÷ 9 + 4 × (15 - 12)
  6. 100 - (25 + 15) ÷ 5 + 8
  7. (4 × 6 - 10) + (3² + 1)
  8. 48 ÷ (12 - 4) × 2 + 6
  9. (20 + 12) ÷ 4 - 3 × (8 - 6)
  10. 7² - (16 ÷ 2 + 5) × 3

These problems encompass a range of operations and complexities, providing ample opportunity to apply the order of operations. As you work through them, pay close attention to each step, ensuring you're adhering to PEMDAS/BODMAS. Don't hesitate to refer back to the explanations and examples provided earlier in this article if you encounter any difficulties.

For each problem, break it down into smaller, manageable steps. Identify the operations within parentheses first, then exponents, followed by multiplication and division (from left to right), and finally addition and subtraction (from left to right). Show your work clearly and neatly, as this will help you track your progress and identify any errors you might make.

After completing the problems, consider checking your answers using a calculator or online equation solver. This will not only verify your results but also provide an opportunity to learn from any mistakes you might have made. Remember, errors are a natural part of the learning process, and analyzing them can lead to a deeper understanding of the concepts.

By diligently working through these practice problems, you'll sharpen your skills in evaluating expressions and gain the confidence to tackle more complex mathematical challenges. Embrace the process of learning, and enjoy the satisfaction of mastering this fundamental mathematical skill.

Conclusion

In conclusion, evaluating the expression (64 ÷ 8 + 2 - 10) + (9 × 9 + 19) requires a systematic approach rooted in the order of operations. By meticulously following PEMDAS/BODMAS, we can break down the expression into manageable steps, ensuring accuracy at each stage. We first simplified the expressions within the parentheses, performing division, multiplication, addition, and subtraction in the correct sequence. This led us to the simplified expression 0 + 100, which ultimately yielded the solution of 100. Understanding and applying the order of operations is not just about solving this particular expression; it's a fundamental skill that empowers us to tackle a wide range of mathematical problems with confidence. By mastering this skill, we lay a solid foundation for further exploration and success in the world of mathematics.