Use the quotient rule to simplify. Assume that all variables represent positive real numbers. \( -\sqrt[3]{\frac{z^{11}}{27 x^{9}}} \) \( -\sqrt[3]{\frac{z^{11}}{27 x^{9}}}=\square \) (Simplify your answer. Use integers or fractions for any numbers in the expression. Type an exact answer, using radicals as needed.)

To apply the quotient rule when simplifying the expression \( -\sqrt[3]{\frac{z^{11}}{27 x^{9}}} \), we start by separating the cube root using the property \( \sqrt[3]{\frac{a}{b}} = \frac{\sqrt[3]{a}}{\sqrt[3]{b}} \). So we have: \[ -\sqrt[3]{\frac{z^{11}}{27 x^{9}}} = -\frac{\sqrt[3]{z^{11}}}{\sqrt[3]{27} \cdot \sqrt[3]{x^{9}}} \] Now we can simplify further: \( \sqrt[3]{z^{11}} = z^{\frac{11}{3}} \) \( \sqrt[3]{27} = 3 \) (since \(27 = 3^3\)) \( \sqrt[3]{x^{9}} = x^{3} \) Putting it all together, we get: \[ -\frac{z^{\frac{11}{3}}}{3x^{3}} \] Thus, the simplified expression is: \[ -\frac{z^{\frac{11}{3}}}{3x^{3}} \]