1. **State the problem:** Simplify the expression $$\left(\frac{x^4}{y^6}\right)^8$$. 2. **Recall the power of a quotient rule:** When raising a quotient to a power, apply the exponent to both numerator and denominator: $$\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$$ 3. **Apply the rule:** $$\left(\frac{x^4}{y^6}\right)^8 = \frac{(x^4)^8}{(y^6)^8}$$ 4. **Use the power of a power rule:** $$ (a^m)^n = a^{m \times n} $$ 5. **Calculate the exponents:** $$ (x^4)^8 = x^{4 \times 8} = x^{32} $$ $$ (y^6)^8 = y^{6 \times 8} = y^{48} $$ 6. **Write the simplified expression:** $$ \frac{x^{32}}{y^{48}} $$ 7. **Check the options:** The expression matches the first option: $$\frac{x^{32}}{y^{48}}$$. **Final answer:** $$\frac{x^{32}}{y^{48}}$$