1. **State the problem:** Simplify the expression $$\left(\frac{2xy^{5}}{z^{2}}\right)^3 \cdot \left(\frac{x z^{3}}{y}\right)^4$$. 2. **Recall the power of a quotient rule:** For any expression $$\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$$. 3. **Apply the power to each term inside the parentheses:** $$\left(\frac{2xy^{5}}{z^{2}}\right)^3 = \frac{(2)^3 (x)^3 (y^{5})^3}{(z^{2})^3} = \frac{8 x^{3} y^{15}}{z^{6}}$$ $$\left(\frac{x z^{3}}{y}\right)^4 = \frac{(x)^4 (z^{3})^4}{(y)^4} = \frac{x^{4} z^{12}}{y^{4}}$$ 4. **Multiply the two results:** $$\frac{8 x^{3} y^{15}}{z^{6}} \cdot \frac{x^{4} z^{12}}{y^{4}} = \frac{8 x^{3} y^{15} x^{4} z^{12}}{z^{6} y^{4}}$$ 5. **Combine like bases by adding/subtracting exponents:** $$= 8 x^{3+4} y^{15-4} z^{12-6} = 8 x^{7} y^{11} z^{6}$$ 6. **Final simplified expression:** $$8 x^{7} y^{11} z^{6}$$