Simplifying Radicals Finding The Quotient Of 3√8 Divided By 4√6

In the realm of mathematics, simplifying expressions involving radicals is a fundamental skill. This article delves into the process of finding the quotient of $\frac{3 \sqrt{8}}{4 \sqrt{6}}$, providing a step-by-step solution and a thorough explanation of the underlying principles. We will explore the properties of radicals, the techniques for rationalizing denominators, and the art of simplifying mathematical expressions. By the end of this exploration, you'll not only grasp the solution to this specific problem but also gain a deeper understanding of how to tackle similar mathematical challenges. The options provided are A. $\frac{12 \sqrt{2}-6 \sqrt{3}}{5}$, B. $\frac{3 \sqrt{6}-4 \sqrt{3}}{24}$, C. $\frac{\sqrt{3}}{12}$, and D. $\frac{\sqrt{3}}{2}$. Our journey will involve breaking down the radicals, simplifying the fraction, and arriving at the correct answer through a series of logical steps. So, let's embark on this mathematical adventure together!

Breaking Down the Radicals

To effectively simplify the given expression, $\frac{3 \sqrt{8}}{4 \sqrt{6}}$, our initial focus should be on breaking down the radicals, specifically $\sqrt{8}$ and $\sqrt{6}$, into their simplest forms. This involves identifying perfect square factors within the radicands (the numbers inside the square roots). For $\sqrt{8}$, we can express 8 as a product of 4 and 2, where 4 is a perfect square (2 x 2). Thus, $\sqrt{8}$ can be rewritten as $\sqrt{4 \times 2}$. Applying the property of radicals that states $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$, we can further simplify this to $\sqrt{4} \times \sqrt{2}$. Since $\sqrt{4}$ is equal to 2, we have $\sqrt{8} = 2\sqrt{2}$. Now, let's turn our attention to $\sqrt{6}$. The number 6 can be factored into 2 and 3, neither of which are perfect squares. Therefore, $\sqrt{6}$ remains in its simplest form. By simplifying $\sqrt{8}$ to $2\sqrt{2}$, we've taken the first crucial step in unraveling the original expression. This simplification allows us to rewrite the original quotient as $\frac{3 \times 2\sqrt{2}}{4 \sqrt{6}}$, which is $\frac{6\sqrt{2}}{4\sqrt{6}}$. This transformation sets the stage for further simplification and ultimately leads us closer to the final answer. Understanding how to decompose radicals into their simplest forms is not just a useful skill for this problem, but a fundamental technique in algebra and beyond. The ability to recognize and extract perfect square factors from radicands is essential for simplifying expressions, solving equations, and performing various mathematical operations with radicals. In the next section, we'll explore how to further simplify this expression by rationalizing the denominator, a critical step in obtaining the final quotient.

Simplifying the Expression and Rationalizing the Denominator

Having simplified $\sqrt{8}$ to $2\sqrt{2}$, our expression now stands as $\frac{6\sqrt{2}}{4\sqrt{6}}$. The next logical step in simplifying this quotient is to rationalize the denominator. Rationalizing the denominator means eliminating the radical from the denominator of a fraction. This is typically achieved by multiplying both the numerator and the denominator by a suitable radical expression that will result in a rational number in the denominator. In this case, the denominator is $4\sqrt{6}$, so we will multiply both the numerator and the denominator by $\sqrt{6}$. This gives us: $\frac{6\sqrt{2}}{4\sqrt{6}} \times \frac{\sqrt{6}}{\sqrt{6}} = \frac{6\sqrt{2} \times \sqrt{6}}{4 \times 6}$. Now, let's simplify the numerator. We have $6\sqrt{2} \times \sqrt{6} = 6\sqrt{2 \times 6} = 6\sqrt{12}$. We can further simplify $\sqrt{12}$ by recognizing that 12 can be factored into $4 \times 3$, where 4 is a perfect square. Thus, $\sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3}$. Substituting this back into our expression, we get $6\sqrt{12} = 6 \times 2\sqrt{3} = 12\sqrt{3}$. Now, let's simplify the denominator. We have $4 \times 6 = 24$. So, our expression becomes: $\frac{12\sqrt{3}}{24}$. Finally, we can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 12. This yields: $\frac{12\sqrt{3}}{24} = \frac{\sqrt{3}}{2}$. Therefore, the simplified form of the given quotient is $\frac{\sqrt{3}}{2}$. This process of rationalizing the denominator not only simplifies the expression but also makes it easier to compare with other expressions and perform further calculations. By multiplying the numerator and denominator by $\sqrt{6}$, we effectively transformed the denominator into a rational number, eliminating the radical. This technique is widely used in mathematics to simplify expressions and make them more manageable.

Comparing with the Given Options

After meticulously simplifying the expression $\frac{3 \sqrt{8}}{4 \sqrt{6}}$, we arrived at the simplified form of $\frac{\sqrt{3}}{2}$. Now, the crucial step is to compare this result with the options provided to identify the correct answer. The options given were: A. $\frac{12 \sqrt{2}-6 \sqrt{3}}{5}$, B. $\frac{3 \sqrt{6}-4 \sqrt{3}}{24}$, C. $\frac{\sqrt{3}}{12}$, and D. $\frac{\sqrt{3}}{2}$. By direct comparison, it's evident that our simplified result, $\frac{\sqrt{3}}{2}$, perfectly matches option D. This confirms that option D is the correct answer to the question. The other options can be ruled out as they do not match our simplified result. Option A involves a more complex expression with both $\sqrt{2}$ and $\sqrt{3}$ terms in the numerator and a denominator of 5. Option B also has a complex numerator with $\sqrt{6}$ and $\sqrt{3}$ terms, and a denominator of 24. Option C has the correct radical term in the numerator but has a denominator of 12, which is incorrect. The process of comparing our simplified result with the given options highlights the importance of careful simplification and attention to detail in mathematics. Each step in the simplification process, from breaking down the radicals to rationalizing the denominator, plays a crucial role in arriving at the correct answer. By systematically working through the problem and comparing our result with the options, we can confidently identify the correct solution. This exercise not only demonstrates the solution to this particular problem but also reinforces the importance of accuracy and precision in mathematical problem-solving.

In conclusion, we have successfully determined the quotient of $\frac{3 \sqrt{8}}{4 \sqrt{6}}$ through a series of methodical steps. Our journey began with breaking down the radicals, simplifying $\sqrt{8}$ to $2\sqrt{2}$. We then simplified the expression to $\frac{6\sqrt{2}}{4\sqrt{6}}$ and proceeded to rationalize the denominator by multiplying both the numerator and the denominator by $\sqrt{6}$. This crucial step led us to $\frac{12\sqrt{3}}{24}$, which we further simplified to $\frac{\sqrt{3}}{2}$. Finally, by comparing our simplified result with the given options, we confidently identified option D, $\frac{\sqrt{3}}{2}$, as the correct answer. This exploration underscores the significance of mastering fundamental mathematical techniques such as simplifying radicals and rationalizing denominators. These skills are not only essential for solving specific problems but also form the building blocks for more advanced mathematical concepts. The ability to manipulate radical expressions with ease and accuracy is a valuable asset in various fields, including algebra, calculus, and physics. Moreover, this exercise highlights the importance of a systematic approach to problem-solving in mathematics. By breaking down a complex problem into smaller, manageable steps, we can navigate through the solution process with clarity and precision. Each step, from simplifying radicals to rationalizing denominators, contributes to the overall solution and reinforces our understanding of mathematical principles. Therefore, the process of solving this quotient serves as a valuable learning experience, reinforcing both specific mathematical skills and general problem-solving strategies. The correct answer is D. $\frac{\sqrt{3}}{2}$.