1. **Problem Statement:** We have a triangle with angles 30° and 60°, and the side opposite the 60° angle is given as $\sqrt{10}$. We need to find the length of side $x$, which is opposite the 30° angle, in simplest radical form with a rational denominator. 2. **Relevant Formula:** In 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 Known Side:** The side opposite 60° is $\sqrt{10}$, so: $$x\sqrt{3} = \sqrt{10}$$ 4. **Solve for $x$:** $$x = \frac{\sqrt{10}}{\sqrt{3}}$$ 5. **Rationalize the denominator:** $$x = \frac{\sqrt{10}}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{10} \times \sqrt{3}}{3} = \frac{\sqrt{30}}{3}$$ 6. **Final answer:** $$x = \frac{\sqrt{30}}{3}$$ This is the length of side $x$ in simplest radical form with a rational denominator.