Simplifying $(\sqrt{3}-\sqrt{2})(-\sqrt{15}+\sqrt{12})$ A Detailed Guide

In the realm of mathematics, simplifying radical expressions is a fundamental skill. This article delves into the process of simplifying the expression $(\sqrt{3}-\sqrt{2})(-\sqrt{15}+ \sqrt{12})$. We will break down the steps, providing a detailed explanation for each, ensuring a clear understanding of the underlying concepts. This comprehensive guide aims to equip you with the knowledge and techniques to tackle similar problems with confidence. Let's embark on this mathematical journey together, unlocking the secrets of radical simplification and enhancing your problem-solving prowess. We'll explore the intricacies of radicals, focusing on how to manipulate and combine them effectively. This involves understanding the properties of square roots, such as the distributive property and the ability to simplify radicals by factoring out perfect squares. By the end of this exploration, you'll not only be able to simplify this specific expression but also possess a broader understanding of radical simplification techniques applicable to various mathematical scenarios. Remember, practice is key to mastering these skills, so be sure to work through additional examples to solidify your understanding. We'll also touch upon common pitfalls and how to avoid them, ensuring you develop a robust and accurate approach to simplifying radical expressions.

Understanding the Basics of Radicals

Before diving into the simplification process, let's solidify our understanding of radicals. A radical, in its simplest form, is a root of a number. The most common type is the square root, denoted by the symbol $\sqrt{ }$. The square root of a number 'x' is a value that, when multiplied by itself, equals 'x'. For instance, $\sqrt{9} = 3$ because 3 * 3 = 9. Radicals can also represent cube roots, fourth roots, and so on, each indicated by a small number (the index) placed above the radical symbol (e.g., $\sqrt[3]{ }$ for cube root). Understanding these basic concepts is crucial for simplifying complex radical expressions. When dealing with radicals, it's important to remember that we can only combine like terms, meaning terms that have the same radical part. For example, we can combine $2\sqrt{3}$ and $5\sqrt{3}$ because they both have $\sqrt{3}$, but we cannot directly combine $2\sqrt{3}$ and $5\sqrt{2}$ because they have different radical parts. The process of simplifying radicals often involves identifying and factoring out perfect squares (or cubes, etc.) from the radicand (the number inside the radical). This allows us to reduce the radical to its simplest form. Furthermore, understanding the properties of radicals, such as the product and quotient rules, is essential for manipulating and simplifying complex expressions. These rules allow us to break down radicals into smaller, more manageable parts, making the simplification process more straightforward. By mastering these fundamental concepts, you'll be well-equipped to tackle a wide range of radical simplification problems.

Breaking Down the Expression $(\sqrt{3}-\sqrt{2})(-\sqrt{15}+\sqrt{12})$

The given expression is $(\sqrt{3}-\sqrt{2})(-\sqrt{15}+\sqrt{12})$. To simplify this, we will use the distributive property (also known as the FOIL method). This involves multiplying each term in the first parenthesis by each term in the second parenthesis. Let's break it down step by step. First, we multiply $\sqrt{3}$ by both terms in the second parenthesis: $\sqrt{3} * -\sqrt{15}$ and $\sqrt{3} * \sqrt{12}$. Next, we multiply $-\sqrt{2}$ by both terms in the second parenthesis: $-\sqrt{2} * -\sqrt{15}$ and $-\sqrt{2} * \sqrt{12}$. This gives us four terms: $-\sqrt{3*15} + \sqrt{3*12} + \sqrt{2*15} - \sqrt{2*12}$. Now, we need to simplify each of these terms individually. This involves looking for perfect square factors within the radicals. Remember, the goal is to express each radical in its simplest form by factoring out any perfect squares. This process not only makes the expression more manageable but also allows us to identify any like terms that can be combined later on. By carefully applying the distributive property and then simplifying each resulting term, we can systematically reduce the complexity of the original expression. This step-by-step approach ensures accuracy and helps in understanding the underlying principles of radical simplification. Let's continue this journey and unravel the intricacies of this expression.

Simplifying Individual Terms: A Step-by-Step Guide

Now, let's simplify each term obtained in the previous step: $-\sqrt{3*15} + \sqrt{3*12} + \sqrt{2*15} - \sqrt{2*12}$.

• Term 1: $-\sqrt{3*15} = -\sqrt{45}$. We can simplify $\sqrt{45}$ by factoring out the perfect square 9: $-\sqrt{45} = -\sqrt{9*5} = -3\sqrt{5}$.
• Term 2: $\sqrt{3*12} = \sqrt{36}$. Since 36 is a perfect square, $\sqrt{36} = 6$.
• Term 3: $\sqrt{2*15} = \sqrt{30}$. The number 30 has no perfect square factors other than 1, so $\sqrt{30}$ remains as is.
• Term 4: $-\sqrt{2*12} = -\sqrt{24}$. We can simplify $\sqrt{24}$ by factoring out the perfect square 4: $-\sqrt{24} = -\sqrt{4*6} = -2\sqrt{6}$.

So, the simplified terms are: $-3\sqrt{5} + 6 + \sqrt{30} - 2\sqrt{6}$. This process of simplifying each term individually is crucial for arriving at the final answer. By identifying and factoring out perfect squares, we reduce the radicals to their simplest forms, making the expression easier to manage. This step also highlights the importance of understanding the prime factorization of numbers within the radicals. The ability to quickly identify perfect square factors is a valuable skill in simplifying radical expressions. Furthermore, this step-by-step approach minimizes the chances of errors and ensures a clear understanding of the simplification process. As we move forward, we will combine these simplified terms to arrive at the final answer. Remember, the key is to break down the problem into smaller, more manageable parts, making the overall simplification process less daunting.

Combining Like Terms and Final Simplification

After simplifying each term, we have $-3\sqrt{5} + 6 + \sqrt{30} - 2\sqrt{6}$. Now, we need to check if there are any like terms that can be combined. Like terms are those that have the same radical part. In this expression, we have $-3\sqrt{5}$, $\sqrt{30}$, and $-2\sqrt{6}$. None of these have the same radical part, so they cannot be combined. The constant term is 6, which also cannot be combined with the radical terms. Therefore, the expression is already in its simplest form. The final simplified expression is $-3\sqrt{5} + 6 + \sqrt{30} - 2\sqrt{6}$. This final step underscores the importance of recognizing like terms when simplifying radical expressions. Often, after simplifying individual terms, there may be opportunities to further reduce the expression by combining terms with the same radical part. However, in this case, since there are no like terms, the expression remains as is. This highlights the fact that not all radical expressions can be simplified to a single term or a simple numerical value. Sometimes, the simplest form is a combination of radical and constant terms. The ability to recognize when an expression is fully simplified is a crucial skill in mathematics. It demonstrates a thorough understanding of the simplification process and the properties of radicals. In this instance, we have successfully simplified the given expression to its most reduced form, showcasing the application of various radical simplification techniques.

Conclusion: Mastering Radical Simplification

In conclusion, simplifying the expression $(\sqrt{3}-\sqrt{2})(-\sqrt{15}+ \sqrt{12})$ involved a series of steps: applying the distributive property, simplifying individual radical terms by factoring out perfect squares, and finally, checking for like terms to combine. The final simplified form of the expression is $-3\sqrt{5} + 6 + \sqrt{30} - 2\sqrt{6}$. This exercise demonstrates the importance of a systematic approach to simplifying radical expressions. By breaking down the problem into smaller, manageable steps, we can effectively navigate the complexities of radicals and arrive at the correct solution. Mastering these techniques is crucial for success in various areas of mathematics, including algebra, calculus, and beyond. Radical simplification is not just about finding the right answer; it's about developing a deeper understanding of mathematical principles and problem-solving strategies. This process encourages logical thinking, attention to detail, and the ability to apply learned concepts in new situations. As you continue your mathematical journey, remember that practice is key to mastering radical simplification and other essential skills. The more you work with these concepts, the more confident and proficient you will become. Embrace the challenge, explore different approaches, and celebrate the satisfaction of solving complex mathematical problems.