1. **State the problem:** Simplify the expression $$\left( \frac{a^3}{b^6} \right)^7$$. 2. **Recall the power of a quotient rule:** When raising a quotient to a power, apply the exponent to both numerator and denominator: $$\left( \frac{x}{y} \right)^n = \frac{x^n}{y^n}$$. 3. **Apply the rule:** $$\left( \frac{a^3}{b^6} \right)^7 = \frac{(a^3)^7}{(b^6)^7}$$. 4. **Use the power of a power rule:** $$ (x^m)^n = x^{m \times n} $$. 5. **Simplify numerator and denominator:** $$ (a^3)^7 = a^{3 \times 7} = a^{21} $$ $$ (b^6)^7 = b^{6 \times 7} = b^{42} $$ 6. **Write the simplified expression:** $$ \frac{a^{21}}{b^{42}} $$ **Final answer:** $$\boxed{\frac{a^{21}}{b^{42}}}$$