Use the quotient rule to simplify. Assume that all variables represent positive real numbers. \( -\sqrt[3]{\frac{z^{10}}{27 x^{9}}} \) \( -\sqrt[3]{\frac{z^{10}}{27 x^{9}}}=\square \) (Simplify your answer. Use integers or fractions for any numbers in the expression. Type an exact answer, using radicals as needed.)

To simplify the expression \( -\sqrt[3]{\frac{z^{10}}{27 x^{9}}} \), we can apply the quotient rule for radicals, which tells us that the cube root of a fraction can be simplified as follows: \[ -\sqrt[3]{\frac{z^{10}}{27 x^{9}}} = -\frac{\sqrt[3]{z^{10}}}{\sqrt[3]{27} \cdot \sqrt[3]{x^{9}}} \] Now, simplifying each part: 1. The cube root of \( z^{10} \) is \( z^{10/3} \). 2. The cube root of \( 27 \) is \( 3 \). 3. The cube root of \( x^9 \) is \( x^{9/3} = x^{3} \). Putting it together, we have: \[ -\frac{z^{10/3}}{3x^3} \] Thus, the final simplified expression is \[ - \frac{z^{10/3}}{3 x^3} \]