Simplifying The Expression (5 + √(25 + 100)) / (2(4)) A Step-by-Step Guide
In the realm of mathematics, simplifying radical expressions is a fundamental skill that allows us to present mathematical statements in their most concise and understandable form. This article will delve into the step-by-step process of simplifying the expression (5 + √(25 + 100)) / (2(4)), highlighting the key principles and techniques involved. By breaking down the expression into manageable components, we can systematically reduce it to its simplest form, revealing its true value and underlying mathematical structure. This process not only enhances our understanding of the expression itself but also strengthens our ability to manipulate and simplify similar expressions in the future. The ability to simplify radical expressions is crucial in various areas of mathematics, including algebra, calculus, and beyond. Understanding how to efficiently simplify these expressions allows for more accurate calculations and a deeper grasp of the mathematical concepts at play. Mastering this skill will undoubtedly prove beneficial in your mathematical journey, paving the way for tackling more complex problems with confidence and ease. Therefore, let's embark on this journey of simplification, unlocking the hidden beauty and elegance within the expression (5 + √(25 + 100)) / (2(4)). This comprehensive guide will equip you with the necessary tools and knowledge to confidently simplify radical expressions, setting you on a path of mathematical mastery. By following each step carefully and understanding the rationale behind it, you'll not only arrive at the simplified form of the expression but also gain a deeper appreciation for the power and precision of mathematics. So, let's begin this exciting exploration and unveil the simplified form of this intriguing mathematical expression.
Step 1: Simplify the Expression Inside the Square Root
The first crucial step in simplifying this expression involves tackling the components within the square root. The expression inside the square root is 25 + 100. According to the order of operations (PEMDAS/BODMAS), we must perform this addition before addressing the square root itself. Adding 25 and 100 yields 125. Therefore, the expression now becomes (5 + √125) / (2(4)). This seemingly simple step is pivotal, as it consolidates the terms under the radical, preparing us for further simplification. The ability to accurately simplify the expression inside the square root is essential for progressing towards the final solution. By correctly performing this addition, we lay the groundwork for the subsequent steps, ensuring that our calculations remain precise and our simplification process remains efficient. This initial simplification sets the stage for a more manageable expression, allowing us to focus on the remaining operations with clarity and confidence. It is this meticulous attention to detail that ultimately leads to a successful simplification of the original expression. This first step demonstrates the importance of following the correct order of operations and the impact it has on the simplification process. By understanding the significance of each step, we can approach complex expressions with a structured and systematic approach. This foundational skill is crucial for tackling more advanced mathematical problems and solidifies our understanding of fundamental mathematical principles. So, having successfully simplified the expression inside the square root, we are now well-positioned to proceed with the remaining steps and uncover the final simplified form of the expression.
Step 2: Simplify the Square Root
Now that we have simplified the expression inside the square root to 125, the next step is to simplify the square root of 125 (√125). To do this, we need to find the prime factorization of 125. Prime factorization involves breaking down a number into its prime factors, which are numbers that are only divisible by 1 and themselves. The prime factorization of 125 is 5 × 5 × 5, or 5³. We can rewrite √125 as √(5² × 5). Using the property of square roots, √(a × b) = √a × √b, we can separate this into √5² × √5. The square root of 5² is 5, so we have 5√5. Substituting this back into our expression, we now have (5 + 5√5) / (2(4)). Simplifying the square root is a crucial step in making the expression more manageable and easier to understand. By identifying the perfect square factors within the radicand (the number inside the square root), we can extract them and reduce the complexity of the expression. This process not only simplifies the numerical value but also reveals the underlying mathematical structure, making it easier to work with in subsequent calculations. The ability to simplify square roots is a fundamental skill in algebra and is essential for solving equations, simplifying expressions, and performing various mathematical operations. Mastering this technique allows us to express numbers in their most simplified radical form, which is often necessary for achieving accurate and concise results. Therefore, the successful simplification of √125 to 5√5 is a significant step forward in our journey to simplify the entire expression. It demonstrates the power of prime factorization and the properties of square roots in reducing complex mathematical terms to their simplest forms.
Step 3: Simplify the Denominator
Having simplified the numerator's radical component, we now shift our focus to the denominator of the expression. The denominator is given as 2(4), which is a straightforward multiplication operation. Multiplying 2 by 4 yields 8. Therefore, the expression now transforms into (5 + 5√5) / 8. This step, while seemingly simple, is crucial for maintaining the accuracy of our simplification process. The denominator, as the divisor, plays a critical role in determining the overall value of the expression. By simplifying the denominator, we ensure that the division operation is performed correctly, leading to a precise and reliable result. The simplification of the denominator often involves basic arithmetic operations, such as multiplication, division, or addition, depending on the specific form of the denominator. In this case, the multiplication of 2 and 4 is a fundamental operation that directly contributes to the overall simplification of the expression. By correctly performing this multiplication, we pave the way for further simplification steps, such as potentially reducing the fraction or simplifying the expression further. Moreover, simplifying the denominator can also reveal any common factors between the numerator and the denominator, which can then be canceled out to further simplify the expression. This process of simplification is not only about reducing the numerical value but also about revealing the underlying mathematical structure and relationships within the expression. Therefore, the successful simplification of the denominator to 8 is a significant step in our quest to simplify the entire expression. It demonstrates the importance of attending to all parts of the expression, both the numerator and the denominator, to ensure a comprehensive and accurate simplification process.
Step 4: Check for Common Factors and Simplify the Fraction
At this stage, our expression stands as (5 + 5√5) / 8. The final step in simplifying this expression is to check for any common factors between the numerator and the denominator. To do this, we can look at the terms in the numerator, 5 + 5√5. We can see that both terms have a common factor of 5. Factoring out the 5 from the numerator, we get 5(1 + √5). Now our expression is 5(1 + √5) / 8. Examining the simplified numerator and the denominator, we can see that there are no common factors between 5(1 + √5) and 8. The factors of 8 are 1, 2, 4, and 8, none of which divide evenly into 5 or (1 + √5). Since there are no common factors to cancel out, the expression is now in its simplest form. Therefore, the simplified form of the expression (5 + √(25 + 100)) / (2(4)) is 5(1 + √5) / 8. This final step highlights the importance of recognizing and factoring out common factors to achieve the most simplified form of an expression. By identifying the greatest common factor (GCF) and dividing both the numerator and the denominator by it, we can reduce the fraction to its lowest terms. This not only makes the expression more concise but also reveals its fundamental structure, making it easier to understand and manipulate in further calculations. The ability to identify and factor out common factors is a crucial skill in algebra and is essential for simplifying fractions, solving equations, and performing various mathematical operations. Mastering this technique allows us to express numbers and expressions in their most simplified form, which is often necessary for achieving accurate and efficient results. Therefore, the successful simplification of the expression to 5(1 + √5) / 8 demonstrates the power of factorization and the importance of checking for common factors in achieving the final simplified form.
Conclusion
In conclusion, the simplified form of the expression (5 + √(25 + 100)) / (2(4)) is 5(1 + √5) / 8. We arrived at this simplified form by systematically breaking down the expression into manageable steps, adhering to the order of operations, and applying fundamental algebraic principles. The journey of simplification began with simplifying the expression inside the square root, followed by simplifying the square root itself. We then simplified the denominator and, finally, checked for common factors to ensure that the expression was in its most reduced form. Each step in this process was crucial, contributing to the overall accuracy and clarity of the final result. This exercise highlights the importance of a methodical approach to mathematical problem-solving, emphasizing the need to carefully consider each component of the expression and apply the appropriate techniques for simplification. Moreover, it reinforces the significance of understanding fundamental mathematical concepts, such as the order of operations, prime factorization, and the properties of square roots. By mastering these concepts, we can confidently tackle more complex mathematical problems and express them in their simplest and most understandable forms. The ability to simplify expressions is a fundamental skill in mathematics, essential for various applications across different fields. From algebra to calculus and beyond, the ability to manipulate and simplify mathematical expressions is crucial for solving equations, analyzing data, and making informed decisions. Therefore, the successful simplification of the expression (5 + √(25 + 100)) / (2(4)) not only provides us with a specific answer but also strengthens our overall mathematical proficiency and prepares us for future challenges. It is through such exercises that we develop a deeper appreciation for the elegance and power of mathematics, unlocking its potential to solve a wide range of problems and reveal the hidden patterns that govern our world.
