Simplifying $-\sqrt{55}(\sqrt{2}+4 \sqrt{2})$ A Step-by-Step Guide

In the realm of mathematics, expressions involving radicals and algebraic manipulations often present intriguing challenges. This article delves into the intricacies of simplifying the expression $-\sqrt{55}(\sqrt{2}+4 \sqrt{2})$, providing a step-by-step guide to unravel its underlying structure and arrive at a concise solution. We will explore the fundamental principles of radical simplification, the distributive property, and the art of combining like terms, all while adhering to the highest standards of mathematical rigor and clarity. This exploration isn't just about finding an answer; it's about understanding the process, the logic, and the beauty inherent in mathematical problem-solving. By meticulously dissecting this problem, we aim to not only arrive at the correct solution but also to equip readers with a deeper understanding of mathematical concepts and techniques that can be applied to a wide range of similar problems. This comprehensive guide is designed to be accessible to learners of all levels, from those just beginning their journey into algebra to seasoned math enthusiasts looking to refine their skills. We'll break down each step with clear explanations and illustrative examples, ensuring that every reader can follow along and grasp the underlying principles. Understanding the fundamentals of simplifying radical expressions is crucial for success in higher-level mathematics, including calculus, trigonometry, and beyond. This article serves as a valuable resource for anyone looking to master these essential skills.

Simplifying the Expression: A Step-by-Step Approach

To effectively simplify $-\sqrt{55}(\sqrt{2}+4 \sqrt{2})$, we embark on a journey through several key mathematical concepts. Firstly, we address the combination of like terms within the parentheses. Secondly, we employ the distributive property to expand the expression. Thirdly, we delve into the simplification of radicals, leveraging factorization and the properties of square roots. Finally, we present the result in its most simplified form. The beauty of mathematics lies in its systematic approach to problem-solving. Each step is built upon the previous one, creating a logical chain that leads to the solution. By carefully following this chain, we can demystify even the most complex-looking expressions. This step-by-step approach is not just a technique for solving this particular problem; it's a valuable skill that can be applied to a wide range of mathematical challenges. It encourages a structured way of thinking, breaking down complex problems into smaller, more manageable steps. This skill is crucial for success in any field that requires analytical thinking and problem-solving, from engineering and computer science to finance and economics. Mastering this approach will not only help you solve mathematical problems more efficiently but also improve your overall problem-solving abilities in all aspects of life. We'll guide you through each step with clear explanations and examples, ensuring that you understand not just what to do but why you're doing it. This emphasis on understanding the underlying principles is what truly differentiates a good learner from a great one.

Step 1: Combining Like Terms

The initial step in simplifying the expression involves recognizing and combining like terms. Within the parentheses, we have $\sqrt{2}$ and $4\sqrt{2}$. These are considered like terms because they share the same radical component, namely $\sqrt{2}$. Combining them is akin to combining apples and apples; we simply add their coefficients. In this case, the coefficient of the first term is implicitly 1, and the coefficient of the second term is 4. Therefore, adding them yields $(1+4)\sqrt{2} = 5\sqrt{2}$. This seemingly simple step is a cornerstone of algebraic manipulation. Recognizing and combining like terms is essential for simplifying expressions and solving equations. It allows us to reduce the complexity of an expression, making it easier to work with. Ignoring this step can lead to unnecessary complications and make it more difficult to arrive at the correct solution. Think of it as organizing your toolbox before starting a project; by grouping similar tools together, you can easily find what you need and work more efficiently. Similarly, by combining like terms, we organize the expression, making it easier to apply further operations. This foundational step lays the groundwork for the subsequent steps in the simplification process. It's a testament to the power of careful observation and attention to detail in mathematics. A keen eye for patterns and the ability to identify like terms will serve you well in your mathematical journey. This skill is not just limited to algebra; it extends to other areas of mathematics, such as calculus and differential equations, where simplifying expressions is a crucial step in solving complex problems.

Step 2: Applying the Distributive Property

Having simplified the terms within the parentheses, our expression now stands as $-\sqrt{55}(5\sqrt{2})$. To further simplify this, we invoke the distributive property. While it might not be immediately obvious how the distributive property applies here, it's crucial to understand that multiplication is inherently distributive over addition and subtraction. In this specific case, we can consider the expression as a single term, $-\sqrt{55}$, multiplying another single term, $5\sqrt{2}$. The distributive property, in its essence, states that a(b + c) = ab + ac. However, in our scenario, we are simply dealing with the multiplication of two terms. Therefore, we proceed by multiplying the coefficients and the radicals separately. This multiplication results in $-5(\sqrt{55} \cdot \sqrt{2})$. The distributive property is a fundamental principle in algebra, and its understanding is crucial for manipulating and simplifying expressions. It allows us to expand expressions, remove parentheses, and rearrange terms, all while maintaining the equality of the expression. It's a versatile tool that is used extensively in various areas of mathematics, from solving equations to proving theorems. Mastering the distributive property is like learning the alphabet of algebra; it's a building block for more advanced concepts. Without a solid understanding of this property, it's difficult to progress in your mathematical studies. This step demonstrates the power of applying fundamental principles in novel situations. While the distributive property is often associated with expressions involving parentheses and addition or subtraction, its underlying concept applies to multiplication in general. Recognizing this subtle connection is a key aspect of mathematical fluency.

Step 3: Simplifying the Radical

At this stage, our expression is $-5(\sqrt{55} \cdot \sqrt{2})$. The next step involves simplifying the product of the radicals. A key property of radicals states that $\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}$. Applying this property, we can rewrite our expression as $-5\sqrt{55 \cdot 2}$, which simplifies to $-5\sqrt{110}$. Now, we delve into simplifying the radical $\sqrt{110}$. This involves finding the prime factorization of 110. The prime factorization of 110 is $2 \cdot 5 \cdot 11$. Since there are no perfect square factors within 110 (i.e., no factors that appear twice), the radical $\sqrt{110}$ cannot be simplified further. Simplifying radicals is a crucial skill in algebra and beyond. It allows us to express radicals in their most concise form, making them easier to work with and compare. The process involves identifying perfect square factors within the radicand (the number under the radical sign) and extracting their square roots. This skill is essential for solving equations involving radicals, simplifying trigonometric expressions, and performing various other mathematical operations. Understanding the properties of radicals and the process of simplification is vital for mathematical fluency. It's a skill that is repeatedly used throughout mathematics, and mastering it will significantly improve your problem-solving abilities. This step highlights the importance of prime factorization in simplifying radicals. Prime factorization is a fundamental concept in number theory, and it has numerous applications in mathematics, including simplifying radicals, finding the greatest common divisor, and solving Diophantine equations.

Step 4: The Final Simplified Form

Having explored all avenues of simplification, we arrive at the final form of our expression. Since $\sqrt{110}$ cannot be simplified further, our simplified expression is $-5\sqrt{110}$. This represents the most concise and elegant form of the original expression, $-\sqrt{55}(\sqrt{2}+4 \sqrt{2})$. Throughout this process, we've emphasized the importance of each step, from combining like terms to applying the distributive property and simplifying radicals. Each step contributes to the overall simplification, and a thorough understanding of these steps is crucial for success in algebra and beyond. The final simplified form is not just an answer; it's a testament to the power of mathematical manipulation and the elegance of concise expressions. It represents the culmination of a logical process, where each step is carefully executed to arrive at the most simplified form. This process of simplification is not just about finding the answer; it's about developing a deeper understanding of the mathematical concepts involved and honing your problem-solving skills. It's a journey that empowers you to tackle more complex problems with confidence and clarity. This final step underscores the importance of perseverance in problem-solving. Even when it seems like there are no further simplifications possible, it's important to continue exploring and applying the tools and techniques you've learned. Sometimes, the simplest form is not immediately apparent, and it requires a diligent and systematic approach to uncover it.

Conclusion: The Elegance of Mathematical Simplification

In conclusion, we have successfully navigated the simplification of the expression $-\sqrt{55}(\sqrt{2}+4 \sqrt{2})$ to its most concise form, $-5\sqrt{110}$. This journey has highlighted the fundamental principles of mathematical manipulation, including combining like terms, applying the distributive property, and simplifying radicals. The process underscores the power of systematic problem-solving and the elegance of expressing mathematical concepts in their simplest forms. The ability to simplify expressions is not just a mathematical skill; it's a valuable tool for critical thinking and problem-solving in various domains. By breaking down complex problems into smaller, more manageable steps, we can approach challenges with confidence and clarity. Mastering these skills opens doors to a deeper understanding of mathematics and its applications in the real world. It empowers you to tackle more complex problems, explore advanced concepts, and appreciate the beauty and elegance of mathematical reasoning. This exploration has also demonstrated the interconnectedness of mathematical concepts. The distributive property, radical simplification, and prime factorization are not isolated topics; they are interconnected tools that work together to solve problems. Understanding these connections is crucial for developing a holistic view of mathematics and its applications. We encourage you to continue exploring the world of mathematics, embracing the challenges and celebrating the triumphs. The journey of mathematical discovery is a rewarding one, filled with opportunities for growth, learning, and appreciation of the beauty and power of numbers and symbols.