What is \( \frac{\left(x^{2} y^{3}\right)^{\frac{1}{3}}}{\sqrt[3]{x^{2} y}} \) in exponential form?

To simplify the expression \( \frac{\left(x^{2} y^{3}\right)^{\frac{1}{3}}}{\sqrt[3]{x^{2} y}} \) into exponential form, we can first rewrite both the numerator and the denominator using exponent rules. Starting with the numerator: \[ \left(x^{2} y^{3}\right)^{\frac{1}{3}} = x^{2 \cdot \frac{1}{3}} y^{3 \cdot \frac{1}{3}} = x^{\frac{2}{3}} y^{1} \] Now, for the denominator: \[ \sqrt[3]{x^{2} y} = (x^{2} y)^{\frac{1}{3}} = x^{2 \cdot \frac{1}{3}} y^{1 \cdot \frac{1}{3}} = x^{\frac{2}{3}} y^{\frac{1}{3}} \] Now we can substitute these results back into the original expression: \[ \frac{x^{\frac{2}{3}} y^{1}}{x^{\frac{2}{3}} y^{\frac{1}{3}}} \] Next, we apply the properties of exponents. When dividing like bases, we subtract the exponents: \[ = x^{\frac{2}{3} - \frac{2}{3}} y^{1 - \frac{1}{3}} = x^{0} y^{\frac{2}{3}} \] Since \( x^{0} = 1 \), we simplify further: \[ = 1 \cdot y^{\frac{2}{3}} = y^{\frac{2}{3}} \] Thus, the final expression in exponential form is: \[ y^{\frac{2}{3}} \]