1. **State the problem:** Simplify the expression $$\left(\frac{x^3 y^2}{z^4}\right)^2$$. 2. **Recall the power of a quotient rule:** 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. **Apply the rule:** $$\left(\frac{x^3 y^2}{z^4}\right)^2 = \frac{(x^3 y^2)^2}{(z^4)^2}$$. 4. **Use the power of a product rule:** $$ (ab)^n = a^n b^n $$, so $$ (x^3 y^2)^2 = (x^3)^2 (y^2)^2 = x^{3 \times 2} y^{2 \times 2} = x^6 y^4 $$. 5. **Simplify the denominator:** $$ (z^4)^2 = z^{4 \times 2} = z^8 $$. 6. **Write the final simplified expression:** $$ \frac{x^6 y^4}{z^8} $$. This is the simplified form of the original expression.