1. **State the problem:** Simplify the expression $$\left(\sqrt[8]{16x^{12}}\right)^6$$. 2. **Recall the rules:** The eighth root can be written as a fractional exponent: $$\sqrt[8]{a} = a^{\frac{1}{8}}$$. 3. **Rewrite the expression:** $$\left(16x^{12}\right)^{\frac{1}{8} \times 6} = \left(16x^{12}\right)^{\frac{6}{8}} = \left(16x^{12}\right)^{\frac{3}{4}}$$ 4. **Apply the exponent to each factor:** $$16^{\frac{3}{4}} \times \left(x^{12}\right)^{\frac{3}{4}}$$ 5. **Simplify each part:** - For 16, since $$16 = 2^4$$, $$16^{\frac{3}{4}} = \left(2^4\right)^{\frac{3}{4}} = 2^{4 \times \frac{3}{4}} = 2^3 = 8$$ - For $$x^{12}$$, $$\left(x^{12}\right)^{\frac{3}{4}} = x^{12 \times \frac{3}{4}} = x^9$$ 6. **Combine the results:** $$8x^9$$ **Final answer:** $$8x^9$$