Simplifying The Product Of Radicals (√12+√6)(√6-√10) A Step-by-Step Guide

Introduction

In the realm of mathematics, simplifying expressions involving radicals is a fundamental skill. Radicals, often represented by the square root symbol ($\sqrt{}$), can sometimes seem daunting, but with the right techniques and understanding, they can be manipulated and simplified effectively. In this article, we will delve into the process of simplifying the product of radicals, focusing specifically on the expression $(\sqrt{12}+\sqrt{6})(\sqrt{6}-\sqrt{10})$. This problem provides an excellent opportunity to explore various concepts, including the distributive property, simplifying radicals, and combining like terms. This exploration isn't just about finding the final answer; it's about understanding the underlying principles and how they apply to various mathematical problems. By breaking down the expression step-by-step, we can gain a deeper appreciation for the elegance and logic inherent in mathematics. So, let's embark on this mathematical journey and unravel the intricacies of this radical expression. The journey through this expression will not only enhance our problem-solving skills but also deepen our understanding of the fundamental principles that govern mathematical operations involving radicals. Understanding these principles allows us to tackle more complex problems with confidence and precision. We will begin by dissecting the initial expression, understanding each term individually before applying the distributive property to expand the product. This approach ensures that we handle each component with care and precision, minimizing the chances of errors and maximizing our understanding of the process. From there, we will explore simplifying individual radicals, identifying perfect square factors within the radicands, and extracting them to simplify the expression. This step is crucial in reducing the expression to its simplest form, making it easier to comprehend and work with. Finally, we will combine like terms, applying the rules of addition and subtraction to consolidate the expression into a concise and final result. This final step will not only give us the numerical answer but also demonstrate the power of algebraic manipulation in simplifying complex expressions. Throughout this article, we will emphasize clarity and precision, ensuring that each step is explained in detail and that the underlying reasoning is transparent. This approach is essential for fostering a deep understanding of the concepts involved and for building the confidence to tackle similar problems in the future. So, let's begin our exploration of this intriguing mathematical expression and discover the beauty and logic within it.

Applying the Distributive Property

The first step in simplifying the expression $(\sqrt{12}+\sqrt{6})(\sqrt{6}-\sqrt{10})$ involves applying the distributive property. The distributive property is a fundamental concept in algebra that allows us to multiply a sum or difference by another term. In this case, we need to multiply each term in the first parenthesis by each term in the second parenthesis. This process, often remembered by the acronym FOIL (First, Outer, Inner, Last), ensures that we account for all possible products. Applying the distributive property, we get: $(\sqrt{12} * \sqrt{6}) + (\sqrt{12} * -\sqrt{10}) + (\sqrt{6} * \sqrt{6}) + (\sqrt{6} * -\sqrt{10})$. Now, let's break down each of these products individually. The first product, $(\sqrt{12} * \sqrt{6})$, involves multiplying two square roots. According to the properties of radicals, we can multiply the numbers inside the square roots: $\sqrt{12 * 6} = \sqrt{72}$. The second product, $(\sqrt{12} * -\sqrt{10})$, can be simplified similarly: $-\sqrt{12 * 10} = -\sqrt{120}$. Moving on to the third product, $(\sqrt{6} * \sqrt{6})$, we encounter a special case where the square root of a number is multiplied by itself. This results in the number itself: $\sqrt{6} * \sqrt{6} = 6$. Finally, the fourth product, $(\sqrt{6} * -\sqrt{10})$, can be simplified as: $-\sqrt{6 * 10} = -\sqrt{60}$. So, after applying the distributive property, our expression becomes: $\sqrt{72} - \sqrt{120} + 6 - \sqrt{60}$. This is a significant step forward, but we're not done yet. The next step is to simplify the individual radicals by identifying and extracting perfect square factors. This process will help us reduce the radicals to their simplest forms and make the expression easier to manage. The distributive property is not just a mechanical rule; it's a powerful tool that allows us to break down complex expressions into smaller, more manageable parts. By applying it correctly, we can transform an intimidating problem into a series of simpler calculations. This approach is fundamental to problem-solving in mathematics and extends far beyond the realm of radicals. It's a testament to the importance of understanding the underlying principles that govern mathematical operations. As we continue our journey through this problem, we will see how each step builds upon the previous one, demonstrating the interconnectedness of mathematical concepts and the power of systematic problem-solving.

Simplifying the Radicals

After applying the distributive property, our expression is now $\sqrt{72} - \sqrt{120} + 6 - \sqrt{60}$. The next crucial step is to simplify each radical individually. Simplifying radicals involves identifying perfect square factors within the radicand (the number inside the square root symbol) and extracting them. A perfect square is a number that can be obtained by squaring an integer (e.g., 4, 9, 16, 25, etc.). Let's start with $\sqrt{72}$. We need to find the largest perfect square that divides 72. We can break down 72 into its factors: 72 = 2 * 36. Since 36 is a perfect square (6^2 = 36), we can rewrite $\sqrt{72}$ as $\sqrt{36 * 2}$. Using the property of radicals that $\sqrt{a * b} = \sqrt{a} * \sqrt{b}$, we get $\sqrt{36} * \sqrt{2} = 6\sqrt{2}$. Now, let's move on to $\sqrt{120}$. We need to find the largest perfect square that divides 120. The factors of 120 are: 120 = 4 * 30. Here, 4 is a perfect square (2^2 = 4). So, we can rewrite $\sqrt{120}$ as $\sqrt{4 * 30}$. Applying the same property, we get $\sqrt{4} * \sqrt{30} = 2\sqrt{30}$. Next, we have $\sqrt{60}$. The factors of 60 are: 60 = 4 * 15. Again, 4 is a perfect square. So, we can rewrite $\sqrt{60}$ as $\sqrt{4 * 15}$. Applying the property, we get $\sqrt{4} * \sqrt{15} = 2\sqrt{15}$. Now that we've simplified each radical, our expression becomes $6\sqrt{2} - 2\sqrt{30} + 6 - 2\sqrt{15}$. This simplification is a significant step towards obtaining the final answer. By extracting the perfect square factors, we have reduced the radicals to their simplest forms. This makes the expression much easier to work with and allows us to see if there are any further simplifications possible. The process of simplifying radicals is not just a mechanical exercise; it's a way of revealing the underlying structure of the numbers. By identifying and extracting perfect square factors, we are essentially decomposing the numbers into their fundamental components. This process not only simplifies the expression but also deepens our understanding of the relationships between numbers. As we continue our journey through this problem, we will see how this simplification allows us to combine like terms and arrive at the final answer. The ability to simplify radicals is a crucial skill in mathematics, and it is essential for tackling a wide range of problems involving square roots and other radical expressions. Mastering this skill requires a combination of understanding the properties of radicals and the ability to identify perfect square factors. With practice and perseverance, anyone can develop this skill and use it to simplify complex mathematical expressions.

Combining Like Terms

After simplifying the radicals, our expression is now $6\sqrt{2} - 2\sqrt{30} + 6 - 2\sqrt{15}$. The next step is to combine like terms. In this context, like terms are terms that have the same radical part. For example, $3\sqrt{5}$ and $7\sqrt{5}$ are like terms because they both have $\sqrt{5}$. However, $3\sqrt{5}$ and $3\sqrt{7}$ are not like terms because they have different radicals. Looking at our expression, $6\sqrt{2} - 2\sqrt{30} + 6 - 2\sqrt{15}$, we can see that there are no like terms involving radicals. The terms $6\sqrt{2}$, $-2\sqrt{30}$, and $-2\sqrt{15}$ all have different radicals, so they cannot be combined. The only term that doesn't involve a radical is the constant term, 6. Since there are no other constant terms, it cannot be combined with any other term either. Therefore, the expression $6\sqrt{2} - 2\sqrt{30} + 6 - 2\sqrt{15}$ is already in its simplest form. There are no like terms to combine, so we have reached the final simplified expression. This outcome highlights an important aspect of mathematical problem-solving: not all expressions can be simplified to a single term or a simpler form. Sometimes, the most simplified form is the one where all possible operations have been performed, and no further reduction is possible. In this case, we have applied the distributive property, simplified the radicals, and attempted to combine like terms. Since no further simplification is possible, we can confidently state that $6\sqrt{2} - 2\sqrt{30} + 6 - 2\sqrt{15}$ is the final answer. The process of combining like terms is a fundamental skill in algebra and is essential for simplifying expressions and solving equations. It allows us to consolidate terms that share a common factor, making the expression more concise and easier to understand. However, it is crucial to remember that only like terms can be combined. This means that terms must have the same variable part (in the case of algebraic expressions) or the same radical part (in the case of expressions involving radicals). The ability to identify and combine like terms is a key step in simplifying complex expressions and is a skill that will be used repeatedly in higher-level mathematics. As we have seen in this example, sometimes the most simplified form is not necessarily a single term, but rather an expression where all possible simplifications have been carried out. This understanding is crucial for developing a comprehensive approach to problem-solving and for appreciating the nuances of mathematical expressions.

Conclusion

In conclusion, simplifying the expression $(\sqrt{12}+\sqrt{6})(\sqrt{6}-\sqrt{10})$ involved several key steps. First, we applied the distributive property to expand the product. This transformed the expression into a sum of products of radicals. Next, we simplified each radical by identifying and extracting perfect square factors. This step reduced the radicals to their simplest forms, making the expression easier to manage. Finally, we attempted to combine like terms, but we found that there were no like terms to combine. Therefore, the final simplified expression is $6\sqrt{2} - 2\sqrt{30} + 6 - 2\sqrt{15}$. This problem demonstrates the importance of understanding the properties of radicals and the distributive property. It also highlights the systematic approach required for simplifying mathematical expressions. By breaking down the problem into smaller, more manageable steps, we were able to navigate the complexities of the expression and arrive at the final answer. The process of simplifying radical expressions is not just a mechanical exercise; it's a way of developing a deeper understanding of the relationships between numbers and the properties of mathematical operations. It requires a combination of algebraic manipulation skills and a keen eye for identifying patterns and factors. The distributive property is a fundamental tool in algebra, and its application is not limited to radical expressions. It is a versatile technique that can be used to expand products of any algebraic expressions, including polynomials and other complex structures. Mastering the distributive property is essential for success in algebra and beyond. Simplifying radicals is another crucial skill that is used extensively in mathematics. It allows us to reduce complex expressions to their simplest forms, making them easier to work with and understand. The ability to identify perfect square factors and extract them from radicals is a key component of this skill. Combining like terms is a final step in simplifying many mathematical expressions. It allows us to consolidate terms that share a common factor, making the expression more concise and easier to interpret. However, it is important to remember that only like terms can be combined. The problem we have explored in this article provides a comprehensive example of how these skills can be applied to simplify a complex mathematical expression. By following a systematic approach and applying the appropriate techniques, we were able to arrive at the final answer and gain a deeper appreciation for the beauty and logic of mathematics.