1. **State the problem:** We have a right triangle with angles 30°, 60°, and 90°. The side opposite the 60° angle is 4, and we need to find the length of side $x$, which is opposite the 30° angle, in simplest radical form with a rational denominator. 2. **Recall the properties of a 30°-60°-90° triangle:** The sides are in the ratio $1 : \sqrt{3} : 2$, where: - The side opposite 30° is $x$ (shortest side), - The side opposite 60° is $x\sqrt{3}$, - The hypotenuse is $2x$. 3. **Identify the given side:** The side opposite 60° is given as 4, so: $$x\sqrt{3} = 4$$ 4. **Solve for $x$:** $$x = \frac{4}{\sqrt{3}}$$ 5. **Rationalize the denominator:** Multiply numerator and denominator by $\sqrt{3}$: $$x = \frac{4}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{4\sqrt{3}}{3}$$ 6. **Final answer:** The length of side $x$ in simplest radical form with a rational denominator is: $$\boxed{\frac{4\sqrt{3}}{3}}$$