Solving Mathematical Expressions Using Order Of Operations (PEMDAS)

Let's break down this mathematical expression step-by-step to arrive at the correct solution. This question involves the order of operations, commonly remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). It is important to adhere to this order to accurately solve mathematical problems. Understanding and applying the correct order of operations is crucial in mathematics. This ensures consistency and accuracy in calculations. Many real-world situations, from calculating finances to measuring quantities, rely on the correct application of mathematical principles. Let's delve deeper into each step.

First, we tackle the parentheses: (31 - 29) = 2. This simplifies the expression to 35 - 4^3 ÷ 2. Parentheses indicate the operations that need to be performed first, and they group terms together to change the natural order of operations. This step ensures that the subtraction within the parentheses is performed before any other operations. Parentheses are vital in mathematics because they allow us to control the sequence of operations and thus achieve the correct outcome.

Next up are the exponents: 4^3 = 4 * 4 * 4 = 64. Now the expression looks like this: 35 - 64 ÷ 2. An exponent represents repeated multiplication, and it's crucial to calculate these powers before proceeding with other operations. Exponents are used extensively in various fields like science, engineering, and finance to represent growth, decay, and other exponential relationships. Accurately calculating exponents is essential for solving complex problems involving rates and scaling.

Now we move on to division: 64 ÷ 2 = 32. The expression further simplifies to 35 - 32. Division and multiplication hold the same precedence in the order of operations, and we perform them from left to right. Proper handling of division is essential for calculating ratios, proportions, and splitting quantities, which are fundamental operations in various aspects of daily life and professional calculations.

Finally, we perform the subtraction: 35 - 32 = 3. Therefore, the solution to the expression 35 - 4^3 ÷ (31 - 29) is 3. The final step is to apply subtraction, and this is done after all other operations have been completed according to PEMDAS. Subtraction is one of the basic arithmetic operations, and its accurate execution is critical in many mathematical and real-world problems, such as determining differences, remaining amounts, or changes in quantities.

Let's solve the second mathematical expression, again following the order of operations (PEMDAS). This problem combines various arithmetic operations, making it an excellent example of how PEMDAS helps in solving complex equations. A solid understanding of the order of operations is paramount for accurate calculations in diverse areas of study and real-world applications. Understanding the order of operations is a foundational skill in mathematics. It enables us to handle complex calculations systematically and accurately. This is crucial not only in academic settings but also in everyday situations where precise calculations are required. For example, in finance, engineering, or even cooking, following the correct order of operations can significantly impact the outcome.

First, we focus on the parentheses: (9 + 12 - 2^3). Within the parentheses, we address the exponent first: 2^3 = 2 * 2 * 2 = 8. The expression inside the parentheses becomes (9 + 12 - 8). As seen previously, parentheses group operations, dictating the initial steps in our calculation. They are indispensable in structuring mathematical expressions, especially when multiple operations are involved. Parentheses ensure clarity and prevent ambiguity in complex calculations. They are used across all fields of mathematics and its applications.

Next, within the parentheses, we perform the addition and subtraction from left to right: 9 + 12 = 21, and then 21 - 8 = 13. The expression inside the parentheses simplifies to 13. Adding and subtracting numbers is a fundamental arithmetic skill. Addition and subtraction are the backbone of numerical calculations, utilized in almost every aspect of mathematics. Mastering these operations is essential for both basic arithmetic and more advanced mathematical concepts. They form the basis for understanding other operations and complex mathematical procedures.

Now, let's look at the exponent outside the parentheses: 10^2 = 10 * 10 = 100. The expression becomes 100 × 13 ÷ 4. Exponents denote repeated multiplication and are an essential part of mathematical notation. Calculating exponents accurately is vital in various mathematical and scientific contexts, such as calculating areas, volumes, and exponential growth. Exponents play a crucial role in algebraic expressions and scientific notation, making them a key concept to understand.

We then perform the multiplication: 100 × 13 = 1300. The expression is now 1300 ÷ 4. Multiplication is another fundamental operation. It represents repeated addition and is indispensable in many calculations. It is used extensively in various mathematical and real-world contexts, from calculating areas and volumes to handling rates and proportions. Multiplication is a cornerstone of mathematics and is vital for both basic and advanced problem-solving.

Finally, we perform the division: 1300 ÷ 4 = 325. Therefore, the solution to the expression 10^2 × (9 + 12 - 2^3) ÷ 4 is 325. Division is the inverse operation of multiplication. It allows us to split quantities into equal parts or determine how many times one number fits into another. Understanding division is essential for solving problems related to ratios, proportions, and fractions. This skill is critical in various applications, from everyday calculations to more advanced mathematical problems.

In this mathematical expression, we continue to apply the order of operations (PEMDAS) to find the solution. This equation tests our ability to follow a specific order when multiple operations are involved. Mastering the order of operations is essential for anyone looking to succeed in mathematics and related fields. Adhering to a systematic approach ensures accuracy and consistency in problem-solving. The order of operations is fundamental in mathematical problem-solving. It ensures that calculations are performed in the correct sequence, leading to accurate results. This systematic approach is vital in various fields, from science and engineering to finance and everyday calculations. Understanding and applying PEMDAS is crucial for anyone aiming to master mathematical concepts.

First, we address the exponent: 10^2 = 10 * 10 = 100. The expression becomes 100 × 9 + 12 - 2^3 ÷ 4. Exponents signify repeated multiplication and are pivotal in expressing mathematical relationships and quantities. Accurate calculation of exponents is crucial in many mathematical and scientific contexts, including calculating growth rates, areas, and volumes. Exponents are a cornerstone of algebra and calculus, making a solid understanding of them indispensable for advanced mathematical studies.

Next, we address the other exponent: 2^3 = 2 * 2 * 2 = 8. The expression now looks like this: 100 × 9 + 12 - 8 ÷ 4. This step further simplifies the expression, preparing it for subsequent operations according to PEMDAS. As previously mentioned, exponents are a core part of mathematical notation. They're not just about simplifying calculations; they're also used to express very large and very small numbers in scientific notation, and they're fundamental in many scientific and engineering formulas.

Then, we perform the multiplication: 100 × 9 = 900. The expression simplifies to 900 + 12 - 8 ÷ 4. This highlights the importance of multiplication in the order of operations. Multiplication is a critical operation. It’s a fundamental building block in various calculations, from determining the total cost of multiple items to calculating areas and volumes. Mastery of multiplication is crucial for both basic arithmetic and more advanced mathematical problem-solving.

Following the multiplication, we perform the division: 8 ÷ 4 = 2. The expression becomes 900 + 12 - 2. Here, division is performed before addition and subtraction, aligning with the PEMDAS protocol. Division is essential for sharing, comparing, and finding rates or ratios. Being able to divide correctly is not just useful in math class; it's a skill that is regularly applied in real-life scenarios, such as splitting costs, understanding percentages, and working with scales.

Now we perform addition and subtraction from left to right: 900 + 12 = 912, and then 912 - 2 = 910. Therefore, the solution to the expression 10^2 × 9 + 12 - 2^3 ÷ 4 is 910. Addition and subtraction are the fundamental operations that we use daily. They are essential for calculating totals, differences, and changes. Proficiency in these operations is key to managing finances, measuring quantities, and understanding various mathematical concepts. Mastering these basic operations provides a foundation for tackling more complex mathematical challenges.

This mathematical problem requires us to meticulously follow the order of operations (PEMDAS) to arrive at the correct answer. Complex expressions such as this emphasize the significance of understanding and applying PEMDAS systematically. It's not just about getting the right answer; it's about developing a methodical approach to problem-solving. A deep understanding of the order of operations is more than just a mathematical skill; it's a systematic approach to problem-solving that applies across many disciplines. This skill is essential for accuracy in calculations and for understanding the logical flow of mathematical expressions. Mastering PEMDAS is a cornerstone of mathematical proficiency and is invaluable in both academic and real-world scenarios.

First, let's tackle the parentheses: (12 - 2) = 10. This simplifies the expression to 10^2 × 9 + 10^3 ÷ 4. Parentheses serve to group terms and operations, indicating what should be calculated first. This grouping is essential for maintaining the correct sequence of calculations. Parentheses are a key tool in writing mathematical expressions, allowing us to express complex relationships clearly and unambiguously.

Next, we address the exponents: 10^2 = 10 * 10 = 100 and 10^3 = 10 * 10 * 10 = 1000. The expression becomes 100 × 9 + 1000 ÷ 4. Calculating exponents correctly is crucial for solving mathematical problems. Exponents are fundamental in expressing powers and growth rates. Proficiency in dealing with exponents is critical in various fields, including science, engineering, and finance, where exponential relationships are commonly encountered.

Now we perform the multiplication: 100 × 9 = 900. The expression is now 900 + 1000 ÷ 4. Multiplication is one of the basic arithmetic operations, and it's vital to perform it before addition and subtraction unless otherwise indicated by parentheses. Multiplication is a fundamental operation. It appears in various forms in everyday calculations. Mastering multiplication is essential for efficient and accurate problem-solving in a wide array of contexts.

Following multiplication, we perform the division: 1000 ÷ 4 = 250. The expression simplifies to 900 + 250. Division is the inverse of multiplication and essential for calculating proportions, ratios, and splitting quantities. Division problems often arise in daily life. From splitting a bill among friends to figuring out unit prices while shopping, strong division skills are invaluable.

Finally, we perform the addition: 900 + 250 = 1150. Therefore, the solution to the expression 10^2 × 9 + (12 - 2)^3 ÷ 4 is 1150. Addition is one of the basic operations that's used. It's essential for putting things together and calculating totals. A good understanding of addition is essential for math proficiency. Addition is fundamental to many aspects of life, from managing a budget to following recipes, so mastering it sets a solid foundation for mathematical confidence.

In this final mathematical expression, we continue our practice of solving using the order of operations (PEMDAS). This expression combines several different operations, providing a comprehensive review of PEMDAS principles. Mastery of these principles is crucial for success in mathematics and its practical applications. The order of operations is the backbone of consistent and accurate mathematical calculation. This structured approach minimizes errors and ensures that everyone arrives at the same correct answer. It is a vital skill that enhances mathematical precision in academic, professional, and everyday contexts.

First, we address the parentheses: (42 + 3^3 - 7). Within the parentheses, we start with the exponent: 3^3 = 3 * 3 * 3 = 27. The expression inside the parentheses becomes (42 + 27 - 7). Parentheses are a powerful tool. They clarify the order in which operations should be performed. In complex equations, they are indispensable for ensuring correct evaluation. Without parentheses, the expression might be interpreted differently, leading to an incorrect result.

Next, within the parentheses, we perform addition and subtraction from left to right: 42 + 27 = 69, and then 69 - 7 = 62. The expression inside the parentheses simplifies to 62. Addition and subtraction are the cornerstones of basic arithmetic. This is why mastering these operations is fundamental to mathematical proficiency. These skills are continuously used in daily life, making them not only academically valuable but also practically essential.

Now the expression looks like this: 62 - 2 × 5 + 65. Following PEMDAS, we perform the multiplication: 2 × 5 = 10. The expression becomes 62 - 10 + 65. As highlighted earlier, multiplication is a core arithmetic operation with widespread applications. It's used in calculating totals, areas, volumes, and many other mathematical and real-world scenarios. Proficiency in multiplication is critical for efficient problem-solving.

Finally, we perform addition and subtraction from left to right: 62 - 10 = 52, and then 52 + 65 = 117. Therefore, the solution to the expression (42 + 3^3 - 7) - 2 × 5 + 65 is 117. Addition and subtraction are essential operations for calculating sums, differences, and changes. These calculations form the bedrock of everyday mathematical tasks. Proficiency in addition and subtraction is critical for managing personal finances, following recipes, and various other practical applications.

By methodically applying the order of operations (PEMDAS) in each of these examples, we've ensured accuracy and efficiency in solving the expressions. The order of operations is a fundamental rule in mathematics. This is the key to solving complex problems step-by-step. Understanding PEMDAS not only helps in academic math but also in real-life situations. When you need to make precise calculations, following this order prevents errors and ensures that you arrive at the correct answer.