1. **State the problem:** Simplify the expression $$\left( \frac{2x^3 y^2}{x^3 y^7 x^4} \right)^{-3}$$. 2. **Rewrite the expression inside the parentheses:** Combine like terms in the denominator: $$x^3 y^7 x^4 = x^{3+4} y^7 = x^7 y^7$$ So the expression becomes: $$\left( \frac{2x^3 y^2}{x^7 y^7} \right)^{-3}$$ 3. **Simplify the fraction inside the parentheses:** $$\frac{2x^3 y^2}{x^7 y^7} = 2 \cdot \frac{x^3}{x^7} \cdot \frac{y^2}{y^7} = 2 \cdot x^{3-7} \cdot y^{2-7} = 2x^{-4} y^{-5}$$ 4. **Apply the negative exponent outside the parentheses:** $$\left( 2x^{-4} y^{-5} \right)^{-3} = 2^{-3} \cdot (x^{-4})^{-3} \cdot (y^{-5})^{-3}$$ 5. **Simplify each term:** $$2^{-3} = \frac{1}{2^3} = \frac{1}{8}$$ $$(x^{-4})^{-3} = x^{(-4)(-3)} = x^{12}$$ $$(y^{-5})^{-3} = y^{(-5)(-3)} = y^{15}$$ 6. **Combine all terms:** $$\frac{1}{8} x^{12} y^{15} = \frac{x^{12} y^{15}}{8}$$ **Final answer:** $$\frac{x^{12} y^{15}}{8}$$