Equivalent Expression For $\frac{\sqrt[7]{x^2}}{\sqrt[5]{y^3}}$ Explained

This article aims to clarify the process of converting radical expressions into expressions with rational exponents, focusing on the specific example: $\frac{\sqrt[7]{x^2}}{\sqrt[5]{y^3}}$. This type of conversion is fundamental in algebra and calculus, enabling the simplification and manipulation of complex expressions. We will delve into the properties of exponents and radicals, offering a step-by-step solution that not only answers the question but also enhances understanding of the underlying mathematical principles. Understanding these concepts is crucial for anyone studying mathematics, particularly in algebra, precalculus, and calculus courses.

Understanding Radicals and Rational Exponents

To effectively tackle the problem, it's crucial to first understand the relationship between radicals and rational exponents. A radical is a mathematical expression that involves a root, such as a square root, cube root, or nth root. The general form of a radical is $\sqrt[n]{a}$, where n is the index (the root) and a is the radicand (the value under the radical). For example, in $\sqrt[3]{8}$, 3 is the index and 8 is the radicand. On the other hand, a rational exponent is an exponent that is a fraction. For example, $x^{\frac{1}{2}}$ is an expression with a rational exponent. The numerator of the fraction represents the power to which the base is raised, and the denominator represents the index of the root. The core connection between radicals and rational exponents lies in the fact that they are different ways of expressing the same mathematical concept. Specifically, $\sqrt[n]{a}$ is equivalent to $a^{\frac{1}{n}}$. This equivalence is the foundation for converting between radical and exponential forms and it is a key concept to remember. This understanding is pivotal in simplifying and manipulating algebraic expressions, as it allows us to apply the rules of exponents to radicals and vice versa. The ability to convert between radical and exponential forms not only simplifies calculations but also provides a deeper understanding of the structure and properties of mathematical expressions. In essence, mastering this conversion is a fundamental step in advancing one's mathematical proficiency.

Step-by-Step Conversion

Let's break down the conversion process step-by-step, focusing on the given expression: $\frac{\sqrt[7]{x^2}}{\sqrt[5]{y^3}}$. The first step is to convert each radical term into its equivalent exponential form. We know that $\sqrt[n]{a^m}$ can be written as $a^{\frac{m}{n}}$. Applying this rule to the numerator, $\sqrt[7]{x^2}$ becomes $x^{\frac{2}{7}}$. Here, the index of the radical (7) becomes the denominator of the exponent, and the power of x (2) becomes the numerator. Similarly, for the denominator, $\sqrt[5]{y^3}$ transforms into $y^{\frac{3}{5}}$. The index of the radical (5) is the denominator, and the power of y (3) is the numerator. Now, we can rewrite the original expression as $\frac{x^{\frac{2}{7}}}{y^{\frac{3}{5}}}$. The next crucial step involves dealing with the fraction. When a term with an exponent is in the denominator, we can move it to the numerator by changing the sign of the exponent. In other words, $\frac{1}{a^n}$ is the same as $a^{-n}$. Applying this rule to our expression, we move $y^{\frac{3}{5}}$ from the denominator to the numerator, changing its exponent from $\frac{3}{5}$ to $-\frac{3}{5}$. Therefore, $\frac{x^{\frac{2}{7}}}{y^{\frac{3}{5}}}$ becomes $x^{\frac{2}{7}} imes y^{-\frac{3}{5}}$. This transformation is a powerful technique in simplifying expressions and is frequently used in various mathematical contexts. By understanding and applying these steps, we can confidently convert radical expressions into equivalent forms with rational exponents, paving the way for further simplification or manipulation as needed.

Identifying the Equivalent Expression

Following the step-by-step conversion, we've successfully transformed the original expression, $\frac{\sqrt[7]{x^2}}{\sqrt[5]{y^3}}$, into its equivalent form with rational exponents. Recall that we first converted the radicals to exponents, resulting in $\frac{x^{\frac{2}{7}}}{y^{\frac{3}{5}}}$. Then, we moved the term in the denominator to the numerator by changing the sign of its exponent, which gave us $x^{\frac{2}{7}} imes y^{-\frac{3}{5}}$. Now, let's examine the given options and identify the one that matches our transformed expression. Option A is \left(x^{\frac{2}{7}} ight)\(y^{-\frac{3}{5}}\}. This option perfectly aligns with our result, $x^{\frac{2}{7}} imes y^{-\frac{3}{5}}$, confirming that it is the equivalent expression. Options B, which is \left(x^{\frac{2}{7}} ight)\(y^{\frac{5}{3}}\}, does not match because the exponent of y is incorrect. It has a positive exponent of $\frac{5}{3}$, whereas our derived expression has a negative exponent of $-\frac{3}{5}$. Therefore, Option B is not the equivalent expression. The correct answer is undoubtedly Option A, as it accurately represents the original radical expression in its equivalent form with rational exponents. This process underscores the importance of understanding and correctly applying the rules of exponents and radicals. By meticulously following each step, we can confidently navigate through mathematical transformations and arrive at the correct solution.

Conclusion

In summary, the expression equivalent to $\frac{\sqrt[7]{x^2}}{\sqrt[5]{y^3}}$ is indeed \left(x^{\frac{2}{7}} ight)\(y^{-\frac{3}{5}}\}, as demonstrated through our step-by-step conversion process. We began by converting the radicals into rational exponents, transforming $\sqrt[7]{x^2}$ into $x^{\frac{2}{7}}$ and $\sqrt[5]{y^3}$ into $y^{\frac{3}{5}}$. This initial step highlights the fundamental relationship between radicals and rational exponents, where the index of the radical becomes the denominator of the fractional exponent. We then addressed the fraction by moving the term from the denominator to the numerator, which involved changing the sign of the exponent. This transformation is a critical technique in simplifying expressions, allowing us to manipulate and combine terms more effectively. The ability to convert between radical and exponential forms is a cornerstone of algebraic manipulation. It not only simplifies calculations but also provides a deeper understanding of the structure and properties of mathematical expressions. Moreover, mastering these conversions is essential for tackling more advanced mathematical concepts in calculus and beyond. The methodical approach we've outlined ensures accuracy and clarity in such transformations. By carefully applying the rules of exponents and radicals, we can confidently navigate complex mathematical problems and arrive at the correct solutions. This example serves as a valuable illustration of how a solid understanding of fundamental principles can empower us to solve challenging problems with precision and insight.