1. **State the problem:** Simplify fully the expression $$\frac{(\sqrt{6})x^{2}}{\sqrt{\frac{96}{x^{10}}}}$$. 2. **Rewrite the denominator:** Recall that $$\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$$, so $$\sqrt{\frac{96}{x^{10}}} = \frac{\sqrt{96}}{\sqrt{x^{10}}} = \frac{\sqrt{96}}{x^{5}}$$ because $$\sqrt{x^{10}} = x^{\frac{10}{2}} = x^{5}$$. 3. **Rewrite the entire expression:** $$\frac{(\sqrt{6})x^{2}}{\frac{\sqrt{96}}{x^{5}}} = (\sqrt{6})x^{2} \times \frac{x^{5}}{\sqrt{96}} = \frac{(\sqrt{6}) x^{2} x^{5}}{\sqrt{96}} = \frac{(\sqrt{6}) x^{7}}{\sqrt{96}}$$. 4. **Simplify the surds:** Note that $$\sqrt{96} = \sqrt{16 \times 6} = \sqrt{16} \times \sqrt{6} = 4 \sqrt{6}$$. 5. **Substitute back:** $$\frac{(\sqrt{6}) x^{7}}{4 \sqrt{6}}$$. 6. **Cancel common surds:** $$\frac{\cancel{\sqrt{6}} x^{7}}{4 \cancel{\sqrt{6}}} = \frac{x^{7}}{4}$$. **Final answer:** $$\boxed{\frac{x^{7}}{4}}$$