1. **State the problem:** Simplify the expression $$\left(\sqrt[8]{16x^{12}}\right)^6$$. 2. **Recall the rule for radicals and exponents:** $$\sqrt[n]{a^m} = a^{\frac{m}{n}}$$ and $$(a^b)^c = a^{bc}$$. 3. **Rewrite the expression using fractional exponents:** $$\left(16x^{12}\right)^{\frac{6}{8}} = \left(16x^{12}\right)^{\frac{3}{4}}$$. 4. **Express 16 as a power of 2:** $$16 = 2^4$$, so $$\left(2^4 x^{12}\right)^{\frac{3}{4}}$$. 5. **Apply the exponent to each factor:** $$2^{4 \times \frac{3}{4}} \times x^{12 \times \frac{3}{4}} = 2^3 \times x^9$$. 6. **Simplify the powers:** $$2^3 = 8$$, so the expression simplifies to $$8x^9$$. **Final answer:** $$8x^9$$.