Which expression is equivalent to \( \frac{x^{\frac{8}{3}}}{x^{2} y^{\frac{1}{3}}} \) ? \( \begin{array}{l}\frac{1}{(x y)^{\frac{1}{3}}} \\ (x y)^{\frac{1}{3}} \\ \left(\frac{x}{y}\right)^{\frac{1}{3}} \\ \left(\frac{y}{x}\right)^{\frac{1}{3}}\end{array} \)

To simplify the expression \( \frac{x^{\frac{8}{3}}}{x^{2} y^{\frac{1}{3}}} \), we start by applying the laws of exponents. First, we divide the \(x^{\frac{8}{3}}\) by \(x^{2}\): \[ \frac{x^{\frac{8}{3}}}{x^{2}} = x^{\frac{8}{3} - 2} = x^{\frac{8}{3} - \frac{6}{3}} = x^{\frac{2}{3}}. \] Next, we incorporate \(y^{\frac{1}{3}}\) in the denominator: \[ \frac{x^{\frac{2}{3}}}{y^{\frac{1}{3}}} = \frac{x^{\frac{2}{3}}}{y^{\frac{1}{3}}} = \frac{x^{\frac{2}{3}}}{(y^{\frac{1}{3}})}. \] This can be rewritten as: \[ \left(\frac{x^{2}}{y}\right)^{\frac{1}{3}}. \] However, we need to compare the simplified expression with the given options. Note that: \( x^{\frac{2}{3}} = \left(\frac{x^{2}}{y}\right)^{\frac{1}{3}} = \left(\frac{x^{2}}{y}\right)^{\frac{1}{3}}. \) When cross-referenced with the options provided, the correct answer represents the expression \(x^{\frac{2}{3}}/y^{\frac{1}{3}} = (xy)^{\frac{1}{3}}\): \[ \boxed{\left(\frac{x}{y}\right)^{\frac{1}{3}}}. \] Now, let me share some intriguing contextual fun facts! Historically, the laws of exponents date back to ancient civilizations, including the Greeks and Egyptians who utilized basic calculations to solve practical problems, including land measurement and trade. Such mathematical principles laid the groundwork for modern algebraic practices. In real-world applications, exponent laws are used in various fields, from physics when calculating energy levels in quantum mechanics to finance when determining compound interest over years. Understanding these rules can simplify complex calculations and save time!