36. Express in simplest form with a rational denominator: \( \frac{4}{\sqrt{12}} \)

To express \( \frac{4}{\sqrt{12}} \) in its simplest form with a rational denominator, follow these steps: 1. **Simplify the Square Root:** \[ \sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3} \] 2. **Rewrite the Original Expression:** \[ \frac{4}{\sqrt{12}} = \frac{4}{2\sqrt{3}} = \frac{2}{\sqrt{3}} \] 3. **Rationalize the Denominator:** Multiply both the numerator and the denominator by \( \sqrt{3} \) to eliminate the square root in the denominator: \[ \frac{2}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{2\sqrt{3}}{3} \] 4. **Final Answer:** \[ \frac{2\sqrt{3}}{3} \] So, the simplest form with a rational denominator is: \[ \boxed{\dfrac{2\,\sqrt{3}}{3}} \]