1. **State the problem:** Simplify the expression $$\left(\frac{zy^4}{4cy}\right)^2$$. 2. **Write the formula and rules:** When raising a fraction to a power, raise both numerator and denominator to that power: $$\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$$. 3. **Simplify inside the parentheses first:** $$\frac{zy^4}{4cy} = \frac{z \cdot y^4}{4c \cdot y}$$ 4. **Cancel common factors:** Since $y$ appears in numerator and denominator, cancel one $y$: $$\frac{z \cdot \cancel{y} y^3}{4c \cdot \cancel{y}} = \frac{z y^3}{4c}$$ 5. **Now raise to the power 2:** $$\left(\frac{z y^3}{4c}\right)^2 = \frac{(z y^3)^2}{(4c)^2}$$ 6. **Apply power to numerator and denominator:** $$\frac{z^2 (y^3)^2}{16 c^2} = \frac{z^2 y^{6}}{16 c^2}$$ 7. **Final simplified expression:** $$\frac{z^2 y^{6}}{16 c^2}$$ This is the simplified form of the original expression.