1. **State the problem:** Simplify the expression $$2 \times x^3 \times \sqrt{2} \times x^3 \div \sqrt[3]{9}$$. 2. **Rewrite the expression:** Group like terms and rewrite radicals with fractional exponents: $$2 \times x^3 \times \sqrt{2} \times x^3 \div \sqrt[3]{9} = 2 \times x^3 \times x^3 \times 2^{\frac{1}{2}} \times 9^{-\frac{1}{3}}$$. 3. **Combine like terms:** Multiply the powers of $x$: $$x^3 \times x^3 = x^{3+3} = x^6$$. 4. **Simplify the constants:** - $2$ remains as is. - $2^{\frac{1}{2}} = \sqrt{2}$. - $9 = 3^2$, so $9^{-\frac{1}{3}} = (3^2)^{-\frac{1}{3}} = 3^{-\frac{2}{3}}$. 5. **Rewrite the expression:** $$2 \times x^6 \times 2^{\frac{1}{2}} \times 3^{-\frac{2}{3}} = 2^{1 + \frac{1}{2}} \times x^6 \times 3^{-\frac{2}{3}} = 2^{\frac{3}{2}} \times x^6 \times 3^{-\frac{2}{3}}$$. 6. **Express the final answer:** $$2^{\frac{3}{2}} x^6 3^{-\frac{2}{3}} = \frac{2^{\frac{3}{2}} x^6}{3^{\frac{2}{3}}}$$. **Final simplified expression:** $$\boxed{\frac{2^{\frac{3}{2}} x^6}{3^{\frac{2}{3}}}}$$