1. **State the problem:** Simplify the expression $\left(3a^{4}b^{6}\right)^{3}$ by applying the exponent to each factor inside the parentheses. 2. **Formula used:** When raising a product to a power, use the rule $\left(xy\right)^n = x^n y^n$. 3. **Apply the exponent:** $$\left(3a^{4}b^{6}\right)^{3} = 3^{3} \cdot \left(a^{4}\right)^{3} \cdot \left(b^{6}\right)^{3}$$ 4. **Simplify each term:** - $3^{3} = 27$ - Use the power of a power rule $\left(x^{m}\right)^{n} = x^{mn}$: $$\left(a^{4}\right)^{3} = a^{4 \times 3} = a^{12}$$ $$\left(b^{6}\right)^{3} = b^{6 \times 3} = b^{18}$$ 5. **Combine all parts:** $$27a^{12}b^{18}$$ **Final answer:** $27a^{12}b^{18}$