Simplifying $3\sqrt{2}(5\sqrt{6}-7\sqrt{3})$ A Step-by-Step Guide

In the realm of mathematics, simplifying expressions involving radicals is a fundamental skill. This article will dissect the expression $3\sqrt{2}(5\sqrt{6}-7\sqrt{3})$, providing a step-by-step guide to arrive at the simplified form. We will explore the underlying principles of radical manipulation, emphasizing the importance of understanding the properties of square roots and the distributive property. By breaking down each step, we aim to make the process accessible and understandable, empowering you to tackle similar problems with confidence. This journey into the world of radicals will not only enhance your mathematical proficiency but also cultivate a deeper appreciation for the elegance and precision inherent in mathematical operations. This exploration will be more than just solving an equation; it will be a journey of understanding the core principles that govern mathematical expressions. We'll delve into the nuances of each step, ensuring that the rationale behind every operation is crystal clear. By the end of this article, you will not only know how to solve this specific problem but also possess the foundational knowledge to approach a wide range of radical simplification challenges. Our goal is to transform the seemingly complex world of radicals into a familiar and manageable landscape, fostering a sense of mastery and enthusiasm for mathematical problem-solving.

Demystifying Radicals: A Comprehensive Approach

Understanding the Basics: The Foundation of Radical Simplification

Before we dive into the specifics of the given expression, it's crucial to establish a firm understanding of the basics of radicals. A radical, denoted by the symbol ${\sqrt{ }}$, represents the root of a number. In the context of this problem, we are primarily dealing with square roots, which seek the value that, when multiplied by itself, equals the number under the radical. For example, ${\sqrt{9} = 3}$ because 3 * 3 = 9. However, many numbers under the radical are not perfect squares, such as 2, 3, and 6, which appear in our expression. These radicals can often be simplified by factoring out perfect squares from the radicand (the number under the radical symbol). This process is essential for combining like terms and arriving at the most simplified form of the expression. Understanding these foundational concepts is not just about memorizing rules; it's about developing an intuitive grasp of how numbers behave under radical operations. This intuitive understanding will empower you to approach more complex problems with creativity and flexibility. It's about seeing the underlying patterns and relationships that govern mathematical expressions, allowing you to navigate the world of radicals with confidence and precision.

The Distributive Property: Our Key to Unlocking the Expression

The distributive property is a fundamental principle in algebra that allows us to multiply a single term by a group of terms within parentheses. In its simplest form, it states that a(b + c) = ab + ac. This property is the key to unraveling the expression $3\sqrt{2}(5\sqrt{6}-7\sqrt{3})$. By distributing the term outside the parentheses ($3\sqrt{2}$) to each term inside the parentheses ($5\sqrt{6}$ and $-7\sqrt{3}$), we can break down the expression into smaller, more manageable parts. This step is crucial because it allows us to eliminate the parentheses and apply the rules of radical multiplication. The distributive property is not just a mathematical tool; it's a way of thinking about how operations interact with each other. It allows us to see complex expressions as a collection of simpler parts, making them easier to understand and manipulate. Mastering the distributive property is essential for success in algebra and beyond, as it forms the basis for many algebraic manipulations and problem-solving strategies. It's a fundamental building block that unlocks the door to more advanced mathematical concepts.

Step-by-Step Solution: A Journey Through Simplification

Step 1: Applying the Distributive Property

The first step in simplifying the expression $3\sqrt{2}(5\sqrt{6}-7\sqrt{3})$ is to apply the distributive property. This involves multiplying $3\sqrt{2}$ by each term inside the parentheses. So, we have:

$3\sqrt{2} * 5\sqrt{6} - 3\sqrt{2} * 7\sqrt{3}$

This step transforms the original expression into a difference of two terms, each involving the product of radicals. This is a crucial step because it allows us to isolate the radical multiplications and apply the rules of radical simplification. The distributive property acts as a bridge, connecting the complex expression to a series of simpler operations. It's a powerful tool that allows us to break down a seemingly daunting problem into smaller, more manageable pieces. By mastering this step, we lay the groundwork for the subsequent simplification processes, paving the way for a clear and efficient solution. It's not just about applying a rule; it's about strategically dismantling the expression to reveal its underlying structure.

Step 2: Multiplying the Radicals

Now, let's focus on simplifying each term obtained in the previous step. We have two terms to simplify: $3\sqrt{2} * 5\sqrt{6}$ and $3\sqrt{2} * 7\sqrt{3}$. When multiplying radicals, we multiply the coefficients (the numbers outside the radical) and the radicands (the numbers inside the radical) separately. Therefore:

• $3\sqrt{2} * 5\sqrt{6} = (3 * 5)\sqrt{2 * 6} = 15\sqrt{12}$
• $3\sqrt{2} * 7\sqrt{3} = (3 * 7)\sqrt{2 * 3} = 21\sqrt{6}$

This step condenses the products of radicals into single radical terms, making the expression easier to manage. It's a direct application of the property ${\sqrt{a} * \sqrt{b} = \sqrt{ab}}$, which is a cornerstone of radical manipulation. By separating the coefficients and radicands, we can apply this property with clarity and precision. This process not only simplifies the expression but also reveals potential opportunities for further simplification by factoring out perfect squares from the radicands. It's a strategic move that sets the stage for the next phase of the solution, where we will delve deeper into the radicands and extract any hidden simplifications. This step is crucial in our journey towards the most concise and elegant form of the expression.

Step 3: Simplifying the Radicands

The next crucial step is to simplify the radicands, which are the numbers under the square root signs. We have $15\sqrt{12}$ and $21\sqrt{6}$. Let's focus on ${\sqrt{12}}$ first. We need to find the largest perfect square that divides 12. The factors of 12 are 1, 2, 3, 4, 6, and 12. Among these, 4 is the largest perfect square (since 4 = 2 * 2). Therefore, we can rewrite ${\sqrt{12}}$ as ${\sqrt{4 * 3}}$. Using the property ${\sqrt{ab} = \sqrt{a} * \sqrt{b}}$, we get ${\sqrt{4 * 3} = \sqrt{4} * \sqrt{3} = 2\sqrt{3}}$. Now, let's consider ${\sqrt{6}}$. The factors of 6 are 1, 2, 3, and 6. There are no perfect square factors other than 1, so ${\sqrt{6}}$ is already in its simplest form. This process of identifying and extracting perfect square factors is at the heart of radical simplification. It allows us to reduce the radicand to its smallest possible value, resulting in a more concise and elegant expression. This step is not just about finding factors; it's about strategically decomposing the radicand to reveal its underlying structure. By mastering this technique, we gain the ability to transform complex radicals into simpler, more manageable forms.

Step 4: Substituting the Simplified Radicals

Now that we've simplified ${\sqrt{12}}$ to $2\sqrt{3}$, we can substitute this back into our expression. Recall that we had $15\sqrt{12} - 21\sqrt{6}$. Replacing ${\sqrt{12}}$ with $2\sqrt{3}$, we get:

$15 * 2\sqrt{3} - 21\sqrt{6} = 30\sqrt{3} - 21\sqrt{6}$

This substitution is a critical step in bringing the solution closer to its final form. By replacing the unsimplified radical with its simplified counterpart, we streamline the expression and make it easier to analyze. This process is not just about plugging in a value; it's about strategically transforming the expression to reveal its underlying structure. It's a way of making the invisible visible, allowing us to see the relationships between the terms more clearly. This step paves the way for the final simplification, where we will assess whether any further operations can be performed. It's a testament to the power of substitution as a tool for simplifying complex mathematical expressions.

Step 5: Final Simplified Form

Looking at the expression $30\sqrt{3} - 21\sqrt{6}$, we need to determine if further simplification is possible. The key here is to examine the radicals: ${\sqrt{3}}$ and ${\sqrt{6}}$. Since ${\sqrt{3}}$ and ${\sqrt{6}}$ are not like radicals (they do not have the same radicand), we cannot combine them directly. Also, ${\sqrt{3}}$ is in its simplest form, as 3 has no perfect square factors other than 1. Similarly, while 6 has factors of 2 and 3, it has no perfect square factors other than 1, meaning ${\sqrt{6}}$ is also in its simplest form. The coefficients, 30 and 21, do have a common factor of 3. However, factoring out 3 would result in $3(10\sqrt{3} - 7\sqrt{6})$, which, while a valid representation, doesn't further simplify the radical terms themselves. Therefore, the expression $30\sqrt{3} - 21\sqrt{6}$ is the final simplified form. This conclusion highlights the importance of recognizing when an expression is in its simplest form. It's not always about performing more operations; it's about recognizing when no further simplification is possible. This step requires a keen eye for detail and a deep understanding of the properties of radicals. It's a testament to the elegance of mathematical simplification, where the goal is not just to arrive at an answer, but to arrive at the most concise and insightful representation of that answer. The final simplified form is not just a solution; it's a statement of mathematical clarity and precision.

Conclusion: Mastering Radical Simplification

In this comprehensive exploration, we have successfully simplified the expression $3\sqrt{2}(5\sqrt{6}-7\sqrt{3})$ to its final form, $30\sqrt{3} - 21\sqrt{6}$. This journey involved a step-by-step application of fundamental mathematical principles, including the distributive property and the simplification of radicals. We began by understanding the basics of radicals and the crucial role of the distributive property in expanding the expression. We then meticulously multiplied the radicals, simplified the radicands by extracting perfect square factors, and substituted the simplified radicals back into the expression. Finally, we recognized that no further simplification was possible, arriving at the most concise representation of the solution. This process underscores the importance of a systematic approach to problem-solving in mathematics. Each step builds upon the previous one, creating a clear and logical path to the solution. Mastering radical simplification is not just about memorizing rules; it's about developing a deep understanding of the underlying concepts and applying them strategically. It's about seeing the connections between different mathematical principles and using them to navigate complex problems with confidence and precision. The skills and insights gained from this exploration will undoubtedly prove invaluable in tackling a wide range of mathematical challenges, empowering you to approach the world of mathematics with a newfound sense of mastery and enthusiasm.