1. **State the problem:** Simplify the expression $$\left(\dfrac{x^6 y^3}{z^9}\right)^{\frac{1}{3}}$$ where all variables are positive real numbers. 2. **Recall the power of a quotient rule:** $$\left(\dfrac{a}{b}\right)^m = \dfrac{a^m}{b^m}$$ and the power of a power rule: $$(a^m)^n = a^{mn}$$. 3. **Apply the power of a quotient rule:** $$\left(\dfrac{x^6 y^3}{z^9}\right)^{\frac{1}{3}} = \dfrac{(x^6 y^3)^{\frac{1}{3}}}{(z^9)^{\frac{1}{3}}}$$ 4. **Apply the power of a product rule:** $$(x^6 y^3)^{\frac{1}{3}} = (x^6)^{\frac{1}{3}} (y^3)^{\frac{1}{3}} = x^{6 \times \frac{1}{3}} y^{3 \times \frac{1}{3}} = x^2 y^1 = x^2 y$$ 5. **Simplify the denominator:** $$(z^9)^{\frac{1}{3}} = z^{9 \times \frac{1}{3}} = z^3$$ 6. **Combine numerator and denominator:** $$\dfrac{x^2 y}{z^3}$$ **Final answer:** $$\dfrac{x^{2} y}{z^{3}}$$