Simplifying Radical Expressions Finding The Equivalent Form Of $\sqrt[4]{\frac{24 X^6 Y}{128 X^4 Y^5}}$

Introduction

In this article, we delve into the intricate process of simplifying radical expressions, specifically focusing on the expression $\sqrt[4]{\frac{24 x^6 y}{128 x^4 y^5}}$. This problem, often encountered in algebra and pre-calculus courses, requires a strong understanding of exponent rules, radical properties, and simplification techniques. Our primary goal is to identify which of the given options (A, B, C, and D) is equivalent to the initial expression, under the assumptions that $x \neq 0$ and $y > 0$. These assumptions are crucial because they ensure that we avoid division by zero and maintain real number results when dealing with even roots. This exploration will not only provide a step-by-step solution but also illuminate the underlying mathematical principles, making it easier to tackle similar problems in the future. Understanding how to simplify radicals is fundamental in various areas of mathematics, including calculus and complex analysis, where expressions often need to be manipulated to more manageable forms for further calculations or analysis. Let's embark on this mathematical journey to unravel the complexities of radical simplification.

Breaking Down the Problem

To effectively simplify the given radical expression, $\sqrt[4]{\frac{24 x^6 y}{128 x^4 y^5}}$, we must first break it down into manageable components. The initial step involves simplifying the fraction inside the radical by reducing the numerical coefficients and applying the quotient rule for exponents. The quotient rule states that when dividing exponential expressions with the same base, we subtract the exponents. This means we will divide 24 by 128 and simplify the expressions involving $x$ and $y$ separately. Once the fraction is simplified, we will apply the properties of radicals to extract any perfect fourth powers from the numerator and the denominator. This process requires us to rewrite the variables with exponents that are multiples of 4, if possible, or identify the largest multiple of 4 that is less than the current exponent. The numerical coefficient will also need to be examined for factors that are perfect fourth powers. By systematically reducing the expression in this manner, we can transform it into its simplest form, which will then allow us to easily identify the correct equivalent expression from the provided options. The ability to break down a complex problem into smaller, more manageable steps is a critical skill in mathematics, as it allows for a more organized and efficient solution process. This approach not only helps in simplifying the expression but also minimizes the chances of making errors along the way.

Step-by-Step Simplification

Our initial expression is $\sqrt[4]{\frac{24 x^6 y}{128 x^4 y^5}}$. The first step involves simplifying the fraction inside the fourth root. Let's begin by reducing the numerical fraction $\frac{24}{128}$. Both 24 and 128 are divisible by 8, so we can simplify the fraction as follows:

$$\frac{24}{128} = \frac{24 \div 8}{128 \div 8} = \frac{3}{16}$$

Next, we simplify the variable terms using the quotient rule for exponents, which states that $\frac{a^m}{a^n} = a^{m-n}$. Applying this rule to the $x$ terms, we have:

$$\frac{x^6}{x^4} = x^{6-4} = x^2$$

Similarly, for the $y$ terms, we have:

$$\frac{y}{y^5} = y^{1-5} = y^{-4} = \frac{1}{y^4}$$

Now, we can rewrite the original expression with these simplified components:

$$\sqrt[4]{\frac{24 x^6 y}{128 x^4 y^5}} = \sqrt[4]{\frac{3 x^2}{16 y^4}}$$

The next step is to separate the radical and simplify further. We can rewrite the expression as:

$$\sqrt[4]{\frac{3 x^2}{16 y^4}} = \frac{\sqrt[4]{3 x^2}}{\sqrt[4]{16 y^4}}$$

We know that $\sqrt[4]{16} = 2$ and $\sqrt[4]{y^4} = y$ (since $y > 0$), so the denominator simplifies to $2y$. Thus, we have:

$$\frac{\sqrt[4]{3 x^2}}{\sqrt[4]{16 y^4}} = \frac{\sqrt[4]{3 x^2}}{2 y}$$

Therefore, the simplified expression is $\frac{\sqrt[4]{3 x^2}}{2 y}$, which matches option D.

Analyzing the Options

After systematically simplifying the given expression $\sqrt[4]{\frac{24 x^6 y}{128 x^4 y^5}}$, we arrived at the simplified form $\frac{\sqrt[4]{3 x^2}}{2 y}$. Now, let's meticulously analyze each of the provided options to confirm our result and understand why the other options are incorrect.

• Option A: $\frac{\sqrt[4]{3}}{2 x^2 y}$

  This option is incorrect because it has $x^2$ in the denominator, while our simplified expression has $\sqrt[4]{x^2}$ in the numerator. The $x^2$ term was correctly simplified within the radical, and moving it to the denominator would be a misapplication of radical properties.

• Option B: $\frac{x(\sqrt[4]{3})}{4 y^2}$

  This option is also incorrect. It incorrectly simplifies the $x$ term and has $y^2$ in the denominator, which is not consistent with our simplified form. The correct simplification involves keeping $x^2$ under the fourth root and having $y$ in the denominator.

• Option C: $\frac{\sqrt[4]{3}}{4 x y^2}$

  This option is incorrect for multiple reasons. It has both $x$ and $y^2$ in the denominator, which do not match our simplified expression. The $x$ term should be part of the radical in the numerator, and the $y$ term should be linear in the denominator.

• Option D: $\frac{\sqrt[4]{3 x^2}}{2 y}$

  This option precisely matches our simplified expression. We correctly simplified the fraction inside the radical, applied the quotient rule for exponents, and extracted the perfect fourth powers. The final form, $\frac{\sqrt[4]{3 x^2}}{2 y}$, aligns perfectly with this option.

By carefully comparing each option with our derived simplified form, we can confidently confirm that option D is the correct answer. This methodical approach of simplification and comparison is crucial for ensuring accuracy in mathematical problem-solving.

Conclusion

In conclusion, through a detailed step-by-step simplification process, we have successfully determined that the expression $\sqrt[4]{\frac{24 x^6 y}{128 x^4 y^5}}$ is equivalent to $\frac{\sqrt[4]{3 x^2}}{2 y}$, which corresponds to option D. This problem highlighted the importance of understanding and applying fundamental concepts such as the quotient rule for exponents, simplifying fractions, and properties of radicals. The ability to break down complex expressions into simpler components is a critical skill in mathematics, enabling us to manipulate and solve problems more efficiently. Moreover, the assumption that $x \neq 0$ and $y > 0$ was crucial to avoid division by zero and ensure that we were dealing with real numbers when taking the fourth root. By systematically working through each step, we not only arrived at the correct answer but also reinforced the underlying mathematical principles. This comprehensive approach ensures a deeper understanding and greater confidence in tackling similar problems in the future. The simplification of radical expressions is a foundational topic in algebra and pre-calculus, with applications extending to more advanced areas of mathematics. Mastering these techniques is therefore essential for continued success in mathematical studies.