Solving (2+5)^2 + (14 / 7) Using PEMDAS: A Step-by-Step Guide

Jul 12, 2025 by ADMIN 62 views

Understanding Order of Operations to Solve Mathematical Expressions

Introduction

In the realm of mathematics, the order of operations is a fundamental concept that dictates the sequence in which mathematical operations should be performed. This ensures consistency and accuracy in calculations, preventing ambiguity and yielding a unique solution for any given expression. The acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) serves as a mnemonic device to remember this order. In this comprehensive guide, we will delve into the intricacies of PEMDAS, applying it to solve the expression $(2+5)^2 + (14 / 7)$ step-by-step. This breakdown will not only provide the solution but also solidify your understanding of how to tackle complex mathematical problems methodically. Mastering the order of operations is crucial for anyone involved in mathematics, whether you're a student, an engineer, or simply someone who enjoys problem-solving. It's the bedrock upon which more advanced mathematical concepts are built, and a firm grasp of PEMDAS will empower you to confidently tackle a wide range of calculations. From simple arithmetic to complex algebraic equations, the principles of PEMDAS remain constant, ensuring that you arrive at the correct answer every time. The importance of this concept cannot be overstated, as it forms the foundation for many other mathematical principles and is used extensively in various fields, including science, engineering, finance, and computer programming. Understanding PEMDAS is not just about following a set of rules; it's about developing a logical and systematic approach to problem-solving. This skill is invaluable in many aspects of life, not just in mathematics. By breaking down complex problems into smaller, manageable steps, you can effectively analyze and solve challenges in any domain. So, let's embark on this journey to master PEMDAS and unlock the power of accurate mathematical computation.

PEMDAS Explained: A Detailed Breakdown

To effectively apply the order of operations, understanding each component of PEMDAS is essential. Let's break down each step in detail:

P - Parentheses (and other grouping symbols): The first step in any mathematical expression is to address operations within parentheses, brackets, or other grouping symbols. These symbols indicate that the enclosed operations should be performed before anything else. This step is crucial because it often dictates the flow of the entire calculation. Grouping symbols can include parentheses (), brackets [], and braces {}. When nested grouping symbols are present (one set inside another), work from the innermost set outwards. Parentheses are not just about simplifying expressions within them; they also serve to clarify the order of operations, especially in complex equations. They ensure that certain operations are performed together as a unit, preventing misinterpretations and errors. For example, in the expression 2 * (3 + 4), the parentheses tell us to add 3 and 4 first, then multiply by 2. Without parentheses, the order would be different, potentially leading to an incorrect result. In essence, parentheses act as a way to prioritize operations, ensuring that the mathematical expression is evaluated in the intended manner. They are a fundamental tool for both simplifying and clarifying mathematical expressions, making them indispensable in any mathematical endeavor.
E - Exponents: Exponents represent repeated multiplication of a base number. After addressing parentheses, we evaluate exponents. This involves raising a number to a power, which means multiplying the base by itself the number of times indicated by the exponent. Exponents are a compact way of expressing repeated multiplication and are a fundamental concept in mathematics. They appear in various areas, from basic arithmetic to advanced calculus. Understanding exponents is crucial for simplifying expressions and solving equations. The exponent indicates how many times the base is multiplied by itself. For example, in the expression 5^3, the base is 5, and the exponent is 3, meaning 5 is multiplied by itself three times (5 * 5 * 5 = 125). Exponents can also be negative or fractional, leading to different types of calculations, such as reciprocals and roots. Negative exponents indicate the reciprocal of the base raised to the positive exponent (e.g., 2^-2 = 1 / 2^2 = 1 / 4). Fractional exponents represent roots (e.g., 9^(1/2) = √9 = 3). Mastering exponents is essential for simplifying algebraic expressions, solving exponential equations, and understanding various mathematical concepts. They provide a powerful tool for representing and manipulating numbers, making them indispensable in both theoretical and practical applications of mathematics.
MD - Multiplication and Division: Multiplication and division hold equal precedence in the order of operations. When both operations appear in an expression, they are performed from left to right. This left-to-right rule ensures consistency and avoids ambiguity in calculations. Multiplication is the process of repeated addition, while division is the inverse operation of multiplication. These operations are fundamental to mathematics and are used extensively in everyday life. Understanding how to perform them correctly is crucial for solving a wide range of problems. The left-to-right rule for multiplication and division is essential because it ensures that the expression is evaluated in a consistent manner. For example, in the expression 10 / 2 * 3, we would first divide 10 by 2, which equals 5, and then multiply 5 by 3, resulting in 15. If we were to multiply first, we would get a different answer, highlighting the importance of following the correct order. This principle applies to both simple and complex expressions, ensuring that the calculations are performed accurately. Multiplication and division are not just isolated operations; they are interconnected and often used together in mathematical expressions. Mastering these operations and the rules that govern them is essential for anyone seeking to excel in mathematics and related fields.
AS - Addition and Subtraction: Similar to multiplication and division, addition and subtraction have equal precedence. They are performed from left to right in the order they appear in the expression. Addition is the process of combining two or more numbers, while subtraction is the inverse operation of addition. These are the most basic arithmetic operations and are essential for performing more complex calculations. Understanding the order in which to perform addition and subtraction is crucial for obtaining the correct result. The left-to-right rule ensures consistency and avoids ambiguity. For example, in the expression 8 - 3 + 2, we would first subtract 3 from 8, which equals 5, and then add 2 to 5, resulting in 7. If we were to add first, we would get a different answer, emphasizing the importance of following the correct order. Addition and subtraction are fundamental operations that are used in a wide range of mathematical contexts. They are the building blocks for more advanced concepts and are essential for solving everyday problems. Mastering these operations and the rules that govern them is crucial for anyone seeking to develop strong mathematical skills. In essence, addition and subtraction are the final steps in the PEMDAS order of operations, but their importance is no less than the other steps. They are the operations that bring the calculation to its final result, and performing them correctly is essential for accuracy.

Step-by-Step Solution of $(2+5)^2 + (14 / 7)$

Now, let's apply PEMDAS to solve the given expression $(2+5)^2 + (14 / 7)$ :

Parentheses: First, we address the operation within the parentheses: $(2+5) = 7$ . This simplifies the expression to $7^2 + (14 / 7)$ . The parentheses serve as a clear indication that this addition must be performed before any other operation. They effectively group the numbers together, treating them as a single entity within the larger expression. Without parentheses, the order of operations might be misinterpreted, leading to an incorrect result. By prioritizing the parentheses, we ensure that the addition is carried out first, setting the stage for the subsequent steps in the calculation. This is a fundamental principle of PEMDAS, and mastering it is crucial for solving more complex mathematical problems. The result of this step, 7, is then carried forward into the next stage of the calculation, replacing the original expression within the parentheses. This simplification is a key aspect of using PEMDAS effectively, as it breaks down the problem into smaller, more manageable parts. The careful and methodical application of this principle is what ultimately leads to the correct solution.
Exponents: Next, we evaluate the exponent: $7^2 = 7 * 7 = 49$ . This further simplifies the expression to $49 + (14 / 7)$ . Exponents represent repeated multiplication, and in this case, the exponent of 2 indicates that the base, 7, is multiplied by itself. Understanding exponents is crucial for various mathematical concepts, from basic arithmetic to advanced calculus. They provide a concise way to express repeated multiplication and are fundamental to many mathematical formulas and equations. The calculation of $7^2$ to 49 is a straightforward application of the definition of exponents. This step is essential in the order of operations because exponents have a higher precedence than multiplication, division, addition, and subtraction. Therefore, they must be addressed before these other operations to ensure the correct result. The simplification achieved by evaluating the exponent allows us to move closer to the final solution of the expression. It reduces the complexity of the problem, making it easier to manage the remaining operations. The value obtained, 49, is then carried forward as we continue to apply the rules of PEMDAS.
Division: Now, we perform the division: $(14 / 7) = 2$ . The expression is now simplified to $49 + 2$ . Division is one of the four basic arithmetic operations and is the inverse of multiplication. In this step, we divide 14 by 7, resulting in 2. This operation is performed before addition because, according to PEMDAS, division and multiplication have higher precedence than addition and subtraction. The order in which these operations are carried out is critical to obtaining the correct answer. If we were to add 49 and 1 before dividing, the result would be incorrect. This highlights the importance of strictly adhering to the order of operations. The result of the division, 2, is then added to 49 in the final step. This step-by-step approach is a key aspect of using PEMDAS effectively. It breaks down the problem into smaller, more manageable parts, making it easier to identify and perform the correct operations in the correct order. The division step brings us closer to the final solution, and the value obtained, 2, plays a crucial role in the ultimate result.
Addition: Finally, we perform the addition: $49 + 2 = 51$ . This is the final step in solving the expression. Addition is the process of combining two or more numbers, and in this case, we are adding 49 and 2. This operation is performed last because, according to PEMDAS, addition and subtraction have the lowest precedence. The result of the addition, 51, is the solution to the original expression. This final step brings the entire calculation to a close, demonstrating the power of following the order of operations to arrive at a precise and accurate answer. The simplicity of the addition belies the complexity of the steps that preceded it. Each step, from addressing the parentheses to evaluating the exponent and performing the division, was essential in preparing the numbers for this final operation. The result, 51, is not just a number; it is the culmination of a systematic and methodical approach to problem-solving. It underscores the importance of PEMDAS as a tool for ensuring accuracy and consistency in mathematical calculations. The journey from the original expression to the final answer illustrates the elegance and effectiveness of the order of operations.

Conclusion

In conclusion, by following the order of operations (PEMDAS), we have successfully solved the expression $(2+5)^2 + (14 / 7)$ , arriving at the solution 51. This exercise highlights the importance of adhering to PEMDAS to ensure accurate calculations in mathematics. The systematic approach ensures that each operation is performed in the correct sequence, preventing errors and leading to the correct result. Mastering PEMDAS is not just about memorizing a set of rules; it's about developing a disciplined and logical approach to problem-solving. This skill is invaluable in various fields, from science and engineering to finance and everyday life. The principles of PEMDAS extend beyond simple arithmetic and are applicable to more complex mathematical expressions and equations. A solid understanding of these principles is essential for anyone pursuing further studies in mathematics or related disciplines. The ability to break down complex problems into smaller, manageable steps is a key benefit of understanding PEMDAS. This approach not only simplifies the calculation process but also reduces the likelihood of making mistakes. The step-by-step solution presented in this guide demonstrates how each operation is performed in the correct order, building upon the previous steps to arrive at the final answer. This methodical approach can be applied to a wide range of mathematical problems, making PEMDAS a versatile and powerful tool. The importance of accuracy in mathematics cannot be overstated, and PEMDAS provides a framework for ensuring that calculations are performed correctly. By following the order of operations, we can avoid ambiguity and arrive at the unique solution for any given expression. This consistency is crucial for both theoretical and practical applications of mathematics. In summary, mastering PEMDAS is an investment in your mathematical skills that will pay dividends throughout your academic and professional life. It is a foundational concept that empowers you to tackle complex problems with confidence and accuracy.