Simplifying Radical Expressions Finding Equivalent Form Of \(\frac{\sqrt{2}}{\sqrt[3]{2}}\)

In the realm of mathematics, simplifying expressions is a fundamental skill. This article delves into the process of finding an equivalent expression for ${\frac{\sqrt{2}}{\sqrt[3]{2}}}$ , a common problem encountered in algebra and pre-calculus. We will explore the underlying principles of exponents and radicals, providing a step-by-step solution to arrive at the correct answer. This exploration will not only help in solving this specific problem but also enhance your understanding of mathematical manipulations involving radicals and fractional exponents. Mastering these concepts is crucial for success in higher-level mathematics, making this a valuable exercise for students and enthusiasts alike. So, let's embark on this mathematical journey to unravel the intricacies of simplifying radical expressions.

Breaking Down the Problem

To effectively tackle the problem, let's first dissect the expression ${\frac{\sqrt{2}}{\sqrt[3]{2}}}$ and understand its components. The expression involves both square roots and cube roots, which are essentially fractional exponents. Understanding fractional exponents is key to simplifying this expression. Recall that ${\sqrt{a} = a^{\frac{1}{2}}}$ and ${\sqrt[3]{a} = a^{\frac{1}{3}}}$ . By converting the radicals into their exponential forms, we can leverage the rules of exponents to simplify the expression. This conversion allows us to manipulate the expression more easily and identify common factors or terms that can be combined. Furthermore, it provides a clearer pathway to finding an equivalent expression. The ability to translate between radical and exponential forms is a cornerstone of algebraic manipulation, and mastering this skill is essential for solving a wide range of mathematical problems. In the subsequent sections, we will apply these concepts to the given expression, demonstrating the power of fractional exponents in simplifying complex mathematical expressions. Therefore, the initial step in solving this problem lies in recognizing and applying the relationship between radicals and exponents.

Converting Radicals to Exponential Form

The first crucial step in simplifying the expression ${\frac{\sqrt{2}}{\sqrt[3]{2}}}$ is to convert the radicals into their equivalent exponential forms. As mentioned earlier, a square root can be represented as an exponent of ${\frac{1}{2}}$ , and a cube root can be represented as an exponent of ${\frac{1}{3}}$ . Thus, ${\sqrt{2}}$ can be rewritten as ${2^{\frac{1}{2}}}$ , and ${\sqrt[3]{2}}$ can be rewritten as ${2^{\frac{1}{3}}}$ . This transformation is not just a notational change; it allows us to apply the rules of exponents, which are much more straightforward to work with than radicals directly. Once we have converted the radicals to exponential form, the expression becomes ${\frac{2^{\frac{1}{2}}}{2^{\frac{1}{3}}}}$ . This form is significantly easier to manipulate because we can now directly apply the quotient rule of exponents. This step highlights the importance of understanding the fundamental relationship between radicals and exponents. The ability to seamlessly convert between these forms is a valuable tool in simplifying and solving algebraic expressions. Therefore, mastering this conversion is essential for anyone seeking to excel in mathematics. The next section will focus on applying the quotient rule of exponents to further simplify the expression.

Applying the Quotient Rule of Exponents

Now that we have the expression in exponential form, ${\frac{2^{\frac{1}{2}}}{2^{\frac{1}{3}}}}$ , we can apply the quotient rule of exponents. This rule states that when dividing exponential expressions with the same base, you subtract the exponents. Mathematically, this is expressed as ${\frac{a^m}{a^n} = a^{m-n}}$ . Applying this rule to our expression, we get ${2^{\frac{1}{2} - \frac{1}{3}}}$ . The next step is to subtract the exponents, which involves finding a common denominator for the fractions ${\frac{1}{2}}$ and ${\frac{1}{3}}$ . The least common denominator for 2 and 3 is 6, so we rewrite the fractions as ${\frac{3}{6}}$ and ${\frac{2}{6}}$ , respectively. Thus, the exponent becomes ${\frac{3}{6} - \frac{2}{6} = \frac{1}{6}}$ . Therefore, the expression simplifies to ${2^{\frac{1}{6}}}$ . This step demonstrates the power of the quotient rule in simplifying expressions with fractional exponents. By applying this rule, we have successfully reduced the complex fraction to a much simpler form. The ability to manipulate exponents is a fundamental skill in algebra, and this example showcases its importance. In the following section, we will convert this exponential form back into radical form to match the given answer choices and complete the simplification process.

Converting Back to Radical Form

Having simplified the expression to ${2^{\frac{1}{6}}}$ , the final step is to convert this exponential form back into radical form. Recall that an expression of the form ${a^{\frac{1}{n}}}$ is equivalent to the ${n}$-th root of ${a}$ , which is written as ${\sqrt[n]{a}}$ . Applying this conversion to our expression, ${2^{\frac{1}{6}}}$ becomes ${\sqrt[6]{2}}$ . This transformation completes the simplification process and allows us to match our result with the given answer choices. The ability to seamlessly transition between exponential and radical forms is a crucial skill in mathematics, as it allows for flexibility in problem-solving. In this case, converting back to radical form provides a clear and concise representation of the simplified expression. This step underscores the importance of understanding the relationship between exponents and radicals and being able to manipulate them effectively. By successfully converting back to radical form, we have arrived at the final answer and can confidently identify the equivalent expression. In the next section, we will summarize the steps taken and highlight the correct answer among the given options.

Identifying the Correct Answer

Having meticulously simplified the expression ${\frac{\sqrt{2}}{\sqrt[3]{2}}}$ , we have arrived at the equivalent form ${\sqrt[6]{2}}$ . Now, we need to identify the correct answer from the given options: A. ${\frac{1}{4}}$ , B. ${\sqrt[6]{2}}$ , C. ${\sqrt{2}}$ , and D. ${\frac{\sqrt{2}}{2}}$ . Comparing our simplified expression with the options, it is clear that option B, ${\sqrt[6]{2}}$ , matches our result. This confirms that our simplification process was accurate and that we have successfully found the equivalent expression. This step is crucial in any mathematical problem-solving process, as it ensures that the final answer is explicitly stated and aligned with the question's requirements. The ability to accurately compare results and identify the correct answer is a testament to a thorough understanding of the problem and the solution process. In this case, the correct answer is ${\sqrt[6]{2}}$ , which demonstrates the power of converting radicals to exponential form, applying the quotient rule of exponents, and converting back to radical form. In the concluding section, we will summarize the key steps and takeaways from this problem-solving exercise.

Conclusion: Key Takeaways

In conclusion, simplifying the expression ${\frac{\sqrt{2}}{\sqrt[3]{2}}}$ involves a series of crucial steps that highlight the fundamental principles of exponents and radicals. The key takeaways from this exercise are as follows: First, understanding the relationship between radicals and fractional exponents is essential. Converting radicals to exponential form allows for easier manipulation using the rules of exponents. Second, the quotient rule of exponents, which states that ${\frac{a^m}{a^n} = a^{m-n}}$ , is a powerful tool for simplifying expressions involving division of terms with the same base. Third, the ability to convert back and forth between exponential and radical forms is crucial for expressing the final answer in the desired format. By applying these principles, we successfully simplified the expression to ${\sqrt[6]{2}}$ , which corresponds to option B. This exercise not only provides a solution to a specific problem but also reinforces the importance of mastering fundamental mathematical concepts. The ability to manipulate exponents and radicals is a valuable skill in algebra and pre-calculus, and this example serves as a practical demonstration of its application. Therefore, understanding and practicing these concepts will undoubtedly enhance your mathematical proficiency.

Final Answer: The final answer is ${\boxed{\sqrt[6]{2}}}$ .