1. The problem asks to find the geometric mean of $\sqrt{6}$ and $\sqrt{216}$. 2. The formula for the geometric mean of two numbers $a$ and $b$ is: $$\text{Geometric Mean} = \sqrt{a \times b}$$ 3. Substitute $a = \sqrt{6}$ and $b = \sqrt{216}$ into the formula: $$\sqrt{\sqrt{6} \times \sqrt{216}}$$ 4. Use the property of square roots that $\sqrt{x} \times \sqrt{y} = \sqrt{xy}$: $$\sqrt{\sqrt{6} \times \sqrt{216}} = \sqrt{\sqrt{6 \times 216}}$$ 5. Calculate the product inside the inner square root: $$6 \times 216 = 1296$$ 6. So the expression becomes: $$\sqrt{\sqrt{1296}}$$ 7. Since $\sqrt{1296} = 36$, we have: $$\sqrt{36}$$ 8. Finally, $\sqrt{36} = 6$. Therefore, the geometric mean of $\sqrt{6}$ and $\sqrt{216}$ is $6$.