1. **State the problem:** Simplify the expression $\left( \frac{2y^4}{4y} \right)^2$. 2. **Write the formula and rules:** When simplifying powers of fractions, use the rule $\left( \frac{a}{b} \right)^n = \frac{a^n}{b^n}$. Also, when dividing powers with the same base, subtract exponents: $y^m / y^n = y^{m-n}$. 3. **Simplify inside the parentheses first:** $$\frac{2y^4}{4y} = \frac{2}{4} \cdot \frac{y^4}{y} = \frac{1}{2} \cdot y^{4-1} = \frac{1}{2} y^3$$ 4. **Rewrite the expression:** $$\left( \frac{1}{2} y^3 \right)^2$$ 5. **Apply the power to both numerator and denominator:** $$\frac{1^2}{2^2} \cdot y^{3 \times 2} = \frac{1}{4} y^6$$ 6. **Final simplified expression:** $$\frac{y^6}{4}$$ This is the simplified form of the original expression.