1. **Problem statement:** We have a square with side length $x$ and half of its diagonal labeled as $\sqrt{6}$. We need to find $x$ in simplest radical form with a rational denominator. 2. **Recall the properties of a square:** - All sides are equal in length. - The diagonal $d$ relates to the side $x$ by the formula $$d = x\sqrt{2}$$ because the diagonal forms a right triangle with two sides of length $x$. 3. **Given:** Half the diagonal is $\sqrt{6}$, so the full diagonal is $$d = 2\times \sqrt{6} = 2\sqrt{6}$$. 4. **Use the diagonal formula:** $$d = x\sqrt{2}$$ Substitute $d = 2\sqrt{6}$: $$2\sqrt{6} = x\sqrt{2}$$ 5. **Solve for $x$:** $$x = \frac{2\sqrt{6}}{\sqrt{2}}$$ 6. **Rationalize the denominator:** $$x = \frac{2\sqrt{6}}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{2\sqrt{6}\sqrt{2}}{2}$$ 7. **Simplify numerator and denominator:** $$x = \cancel{\frac{2}{2}} \sqrt{6 \times 2} = \sqrt{12}$$ 8. **Simplify $\sqrt{12}$:** $$\sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3}$$ **Final answer:** $$x = 2\sqrt{3}$$