1. **State the problem:** Simplify the expression $$\left( \frac{4x^5 y}{16xy^4} \right)^3$$. 2. **Write the formula and rules:** When simplifying powers of fractions, use $$\left( \frac{a}{b} \right)^n = \frac{a^n}{b^n}$$. Also, recall the laws of exponents: - $$\frac{x^m}{x^n} = x^{m-n}$$ - $$ (x^m)^n = x^{mn} $$ 3. **Simplify inside the parentheses first:** $$\frac{4x^5 y}{16xy^4} = \frac{4}{16} \cdot \frac{x^5}{x} \cdot \frac{y}{y^4}$$ 4. **Simplify each part:** $$\frac{4}{16} = \frac{1}{4}$$ $$\frac{x^5}{x} = x^{5-1} = x^4$$ $$\frac{y}{y^4} = y^{1-4} = y^{-3}$$ So inside the parentheses we have: $$\frac{1}{4} x^4 y^{-3}$$ 5. **Rewrite with positive exponents:** $$\frac{x^4}{4 y^3}$$ 6. **Now raise the entire fraction to the 3rd power:** $$\left( \frac{x^4}{4 y^3} \right)^3 = \frac{(x^4)^3}{(4)^3 (y^3)^3} = \frac{x^{12}}{64 y^9}$$ **Final answer:** $$\boxed{\frac{x^{12}}{64 y^9}}$$