1. **Stating the problem:** We want to find the side length of a cube that can be inscribed inside a sphere with radius $\frac{3\sqrt{3}}{2}$.\n\n2. **Formula and explanation:** When a cube is inscribed in a sphere, the sphere passes through all the vertices of the cube. The diagonal of the cube equals the diameter of the sphere.\n\n3. **Key relationship:** Let the side length of the cube be $s$. The space diagonal of the cube is given by $$d = s\sqrt{3}.$$\n\n4. **Sphere diameter:** The diameter of the sphere is twice the radius, so $$D = 2 \times \frac{3\sqrt{3}}{2} = 3\sqrt{3}.$$\n\n5. **Equate diagonal and diameter:** $$s\sqrt{3} = 3\sqrt{3}.$$\n\n6. **Solve for $s$:** Divide both sides by $\sqrt{3}$:\n$$\frac{s\cancel{\sqrt{3}}}{\cancel{\sqrt{3}}} = \frac{3\cancel{\sqrt{3}}}{\cancel{\sqrt{3}}}$$\n$$s = 3.$$\n\n**Final answer:** The side length of the cube is $3$.