1. **State the problem:** Simplify the expression $$\left(\frac{x^{\frac{4}{3}}}{-125 y^{-\frac{9}{3}} z^{\frac{8}{3}}}\right)^{-\frac{1}{3}}$$. 2. **Recall the power of a quotient rule:** $$\left(\frac{a}{b}\right)^m = \frac{a^m}{b^m}$$ and the power of a power rule: $$(a^m)^n = a^{mn}$$. 3. **Apply the negative exponent rule:** $$\left(\frac{a}{b}\right)^{-m} = \left(\frac{b}{a}\right)^m$$. 4. **Rewrite the expression using the negative exponent rule:** $$\left(\frac{x^{\frac{4}{3}}}{-125 y^{-3} z^{\frac{8}{3}}}\right)^{-\frac{1}{3}} = \left(\frac{-125 y^{-3} z^{\frac{8}{3}}}{x^{\frac{4}{3}}}\right)^{\frac{1}{3}}$$ 5. **Apply the power of a quotient rule:** $$= \frac{\left(-125\right)^{\frac{1}{3}} \left(y^{-3}\right)^{\frac{1}{3}} \left(z^{\frac{8}{3}}\right)^{\frac{1}{3}}}{\left(x^{\frac{4}{3}}\right)^{\frac{1}{3}}}$$ 6. **Simplify each term:** - $$\left(-125\right)^{\frac{1}{3}} = -5$$ because $-5 \times -5 \times -5 = -125$. - $$\left(y^{-3}\right)^{\frac{1}{3}} = y^{-3 \times \frac{1}{3}} = y^{-1} = \frac{1}{y}$$. - $$\left(z^{\frac{8}{3}}\right)^{\frac{1}{3}} = z^{\frac{8}{3} \times \frac{1}{3}} = z^{\frac{8}{9}}$$. - $$\left(x^{\frac{4}{3}}\right)^{\frac{1}{3}} = x^{\frac{4}{3} \times \frac{1}{3}} = x^{\frac{4}{9}}$$. 7. **Substitute back:** $$= \frac{-5 \cdot \frac{1}{y} \cdot z^{\frac{8}{9}}}{x^{\frac{4}{9}}} = \frac{-5 z^{\frac{8}{9}}}{y x^{\frac{4}{9}}}$$ **Final answer:** $$\boxed{\frac{-5 z^{\frac{8}{9}}}{y x^{\frac{4}{9}}}}$$