Step-by-Step Guide Simplifying ((20-16)^2 ÷(8)+5)/(3+4)
Introduction
In the realm of mathematics, simplifying complex expressions is a fundamental skill. This article aims to provide a comprehensive, step-by-step guide on how to simplify the mathematical expression \frac{(20-16)^2 \div(8)+5}{(3+4)}. Mastering such simplifications is crucial for students, educators, and anyone involved in fields that require mathematical proficiency. This tutorial breaks down each operation, ensuring clarity and understanding, and reinforces the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). By following this guide, you'll enhance your ability to tackle similar problems and gain a deeper appreciation for mathematical problem-solving. The process involves several key stages, each requiring careful attention to detail. Understanding the hierarchy of mathematical operations is essential to avoid common mistakes and arrive at the correct solution. We will explore how to handle parentheses, exponents, division, addition, and ultimately, simplify the entire fraction. This step-by-step approach not only provides the solution but also elucidates the underlying principles, making it easier to apply these concepts to other mathematical challenges. So, let’s dive into this mathematical journey and demystify the process of simplifying complex expressions.
Understanding the Order of Operations (PEMDAS/BODMAS)
Before we delve into the specifics of simplifying the expression, it's crucial to understand the order of operations. Often remembered by the acronyms PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) or BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction), this set of rules dictates the sequence in which mathematical operations must be performed to achieve the correct result. Understanding PEMDAS/BODMAS is not just a mathematical rule; it's a fundamental principle that ensures consistency and accuracy in mathematical calculations. Without a standardized order, the same expression could yield multiple different answers, leading to confusion and errors. The PEMDAS/BODMAS rule provides a clear roadmap for simplifying any mathematical expression, ensuring that everyone arrives at the same correct answer. First, operations within Parentheses (or Brackets) are addressed. This is often the starting point for simplifying complex expressions, as it isolates specific operations that must be resolved before moving on. Next, Exponents (or Orders) are evaluated. This includes powers and roots, which are essential components of many mathematical problems. Multiplication and Division are performed next, from left to right. These operations have equal priority, so their order is determined by their sequence in the expression. Finally, Addition and Subtraction are carried out, also from left to right. Like multiplication and division, these operations share equal precedence. By adhering to the order of PEMDAS/BODMAS, we can systematically break down even the most daunting expressions into manageable steps. This not only simplifies the calculation process but also minimizes the risk of errors. In the context of our expression \frac{(20-16)^2 \div(8)+5}{(3+4)}, following PEMDAS/BODMAS is paramount. We will first handle the parentheses, then the exponent, followed by division, and finally, addition in the numerator. The denominator will be simplified separately before we combine the results. This methodical approach is the key to successfully navigating complex mathematical expressions.
Step 1 Simplifying the Parentheses
The first step in simplifying the expression \frac(20-16)^2 \div(8)+5}{(3+4)}** is to address the operations within the parentheses. According to the order of operations (PEMDAS/BODMAS), parentheses take precedence over other operations. This means we must calculate the values inside the parentheses before we proceed with any exponents, multiplication, division, addition, or subtraction. Let’s begin with the numerator. Inside the parentheses, we have (20-16). This is a straightforward subtraction problem. Subtracting 16 from 20 gives us 4. So, (20-16) simplifies to 4. Now, let’s move to the denominator. Here, we have (3+4). This is another simple arithmetic operation, an addition. Adding 3 and 4 results in 7. Therefore, (3+4) simplifies to 7. After simplifying the parentheses, our expression now looks like this{7}. We have successfully eliminated the parentheses by performing the operations enclosed within them. This step is crucial because it reduces the complexity of the expression and sets the stage for the subsequent operations. By addressing the parentheses first, we adhere to the fundamental principle of PEMDAS/BODMAS, ensuring that our calculations are performed in the correct order. This methodical approach is essential for accurate mathematical simplification. The next step will involve dealing with the exponent in the numerator, continuing our journey towards a fully simplified expression. Each step we take brings us closer to the final solution, and by understanding the logic behind each operation, we can confidently tackle even more complex mathematical challenges in the future. Simplifying parentheses is a foundational skill, and mastering it is key to success in algebra and beyond.
Step 2 Evaluating the Exponent
Having simplified the expressions within the parentheses, the next step in evaluating \frac4^2 \div(8)+5}{7}** is to address the exponent. According to the order of operations (PEMDAS/BODMAS), exponents come before multiplication, division, addition, and subtraction. In our expression, we have 4^2 in the numerator. This notation means 4 raised to the power of 2, which is equivalent to 4 multiplied by itself. Mathematically, 4^2 = 4 * 4. Performing this multiplication, we find that 4 * 4 = 16. Therefore, 4^2 simplifies to 16. Now, replacing 4^2 with 16 in our expression, we get{7}. The expression is becoming simpler with each step. By addressing the exponent, we have further reduced the complexity of the numerator, making it easier to proceed with the remaining operations. Understanding exponents is crucial in mathematics, as they appear in various contexts, from basic algebra to advanced calculus. Evaluating exponents correctly is essential for accurate calculations. In this case, recognizing that 4^2 means 4 multiplied by itself allowed us to simplify the term efficiently. This step highlights the importance of knowing the definitions and rules associated with different mathematical operations. As we continue to simplify the expression, we are following the PEMDAS/BODMAS guidelines diligently, ensuring that we perform each operation in the correct sequence. The next step will involve handling the division in the numerator, bringing us closer to the final simplified form. By breaking down the problem into manageable steps, we can tackle even complex mathematical expressions with confidence.
Step 3 Performing Division
With the exponent now evaluated, we move on to the next operation in the numerator of our expression: \frac16 \div(8)+5}{7}**. According to the order of operations (PEMDAS/BODMAS), division and multiplication come before addition and subtraction. In this case, we have a division operation{7}. We have successfully performed the division, further simplifying the numerator. This step demonstrates the importance of following the correct order of operations. Had we attempted to add 5 to 16 before dividing by 8, we would have arrived at an incorrect result. Performing division accurately is crucial in mathematics, and understanding its place in the order of operations is essential. Division is the inverse operation of multiplication, and both play a significant role in many mathematical concepts. By simplifying the division in this step, we have made the expression even more manageable. The numerator now consists of a simple addition, which will be our next focus. As we continue to simplify, we are gradually transforming the complex expression into its most basic form. This step-by-step approach not only helps us arrive at the correct solution but also reinforces the importance of methodical problem-solving. Each operation we perform brings us closer to the final answer, and by understanding the principles behind each step, we build a stronger foundation in mathematics.
Step 4 Completing Addition in the Numerator
Having completed the division, the next step in simplifying the expression \frac2+5}{7}** is to perform the addition in the numerator. According to the order of operations (PEMDAS/BODMAS), addition and subtraction are the last operations to be performed, unless parentheses dictate otherwise. In our expression, we have 2 + 5 in the numerator. Adding 2 and 5 together gives us 7. Therefore, 2 + 5 = 7. Now, replacing the addition operation with its result, our expression becomes{7}. We have successfully simplified the numerator to a single number. This step highlights the fundamental nature of addition in mathematics. Completing addition is a basic arithmetic operation, but it is crucial for simplifying expressions and solving equations. In this case, the addition was the final operation in the numerator, allowing us to condense the expression to a simple fraction. The fraction now represents the final step in our simplification process. We have methodically worked through each operation in the numerator, following the order of operations to ensure accuracy. This systematic approach is a key skill in mathematics, enabling us to tackle complex problems by breaking them down into smaller, more manageable steps. The expression \frac{7}{7} is now in a form that is easily recognizable and can be simplified further. The next step will involve simplifying this fraction to its simplest form, leading us to the final solution.
Step 5 Simplifying the Fraction
After performing all the operations in the numerator and denominator, we have arrived at the simplified fraction \frac{7}{7}. The final step in this process is to simplify this fraction to its simplest form. A fraction represents a part of a whole, and simplifying a fraction means expressing it in its lowest terms. In this case, we have a fraction where the numerator and the denominator are the same number, 7. When the numerator and the denominator of a fraction are equal, the fraction is equivalent to 1. This is because any number divided by itself equals 1. Therefore, \frac{7}{7} = 1. We have successfully simplified the fraction to its simplest form. This step demonstrates the fundamental concept of fractions and their simplification. Simplifying fractions is a crucial skill in mathematics, as it allows us to express quantities in the most concise and understandable way. A simplified fraction is easier to work with and interpret. In this particular case, the simplification resulted in a whole number, which is a common outcome when the numerator and denominator are the same. This final step concludes our journey through the simplification of the original expression. We have methodically worked through each operation, following the order of operations (PEMDAS/BODMAS) to ensure accuracy. By breaking down the problem into smaller, manageable steps, we have successfully simplified the complex expression to its simplest form. The final answer, 1, represents the value of the original expression. This exercise highlights the importance of understanding mathematical principles and applying them systematically to solve problems. The ability to simplify expressions is a valuable skill that extends beyond the classroom and into various real-world applications.
Conclusion
In conclusion, we have successfully simplified the mathematical expression \frac{(20-16)^2 \div(8)+5}{(3+4)} by methodically following the order of operations (PEMDAS/BODMAS). This process involved several key steps, each building upon the previous one to gradually reduce the complexity of the expression. We began by simplifying the expressions within the parentheses, which led us to \frac{4^2 \div(8)+5}{7}. Next, we evaluated the exponent, resulting in \frac{16 \div(8)+5}{7}. Then, we performed the division, transforming the expression to \frac{2+5}{7}. Following this, we completed the addition in the numerator, which gave us \frac{7}{7}. Finally, we simplified the fraction to its simplest form, arriving at the answer 1. This step-by-step approach not only allowed us to find the correct solution but also reinforced the importance of understanding and applying mathematical principles. Simplifying mathematical expressions is a fundamental skill that is essential for success in various areas of mathematics and beyond. By mastering the order of operations and breaking down complex problems into smaller, manageable steps, we can confidently tackle even the most challenging expressions. This exercise serves as a valuable example of how methodical problem-solving can lead to accurate and efficient results. The ability to simplify expressions is not just a mathematical skill; it is a critical thinking skill that can be applied in many aspects of life. Whether it's calculating finances, analyzing data, or solving complex problems in science and engineering, the principles of simplification and order of operations are invaluable tools. By practicing and refining these skills, we can enhance our ability to approach problems logically and effectively.
