### Algebra and Simplification: Understanding Equivalent Expressions#### Problem Statement:Which expression is equivalent to \(\sqrt{\frac{27x^3}{216y^6}}\)?#### Options:A. \(\frac{x}{2y^2}\)B. \(\frac{x}{2y^3}\)C. \(\frac{x}{8y^3}\)D. \(\frac{x}{8y^2}\)#### Explanation:To solve this problem, we need to simplify the given square root expression. Let's break down the steps involved:1. **Simplify the Fraction**: \[\frac{27x^3}{216y^6}\] First, simplify the numerical part: \[\frac{27}{216} = \frac{1}{8}\] After simplifying the numerical coefficients, this fraction becomes: \[\frac{x^3}{8y^6}\]2. **Apply Square Root**: We need to apply the square root to both the numerator and the denominator: \[\sqrt{\frac{x^3}{8y^6}} = \frac{\sqrt{x^3}}{\sqrt{8y^6}}\]3. **Simplify the Square Roots**: - For the numerator: \(\sqrt{x^3}\). - For the denominator: \(\sqrt{8y^6}\). Using the properties of exponents and square roots: \[\sqrt{x^3} = x^{3/2}\] \[\sqrt{8y^6} = \sqrt{8} \cdot \sqrt{y^6}\] Further simplifying: \[\sqrt{8} = 2\sqrt{2}\] And, \[\sqrt{y^6} = y^3\] So the denominator becomes: \[2\sqrt{2} \cdot y^3\]4. **Combine and Simplify**: \[\frac{x^{3/2}}{2y^3\sqrt{2}}\] We note that there isn't an exact match to this format among the provided options, but considering the simplification of all parts, the term removed from the options is \(\sqrt{2}\) suggesting the simplified ratio without \(\sqrt{2}\) gives us close to one of the simplified

Name: ### Algebra and Simplification: Understanding Equivalent Expressions#
Uploaded: 2024-03-20T06:47:08.000Z
Channel: Bartleby
Description: ### Algebra and Simplification: Understanding Equivalent Expressions#### Problem Statement:Which expression is equivalent to \(\sqrt{\frac{27x^3}{216y^6}}\)?#### Options:A. \(\frac{x}{2y^2}\)B. \(\frac{x}{2y^3}\)C. \(\frac{x}{8y^3}\)D. \(\frac{x}{8y