7. A regular tetrahedron with edge length $x$ is inside a cube with edge length 6, and the tetrahedron can rotate freely within the cube. The maximum value of $x$ is $\qquad$ .
Answer
As shown in the figure, according to the problem, the largest regular tetrahedron is the one inscribed in the sphere that is inscribed in the cube, with its height being $\frac{4}{3} R=4$. From the relationship between the height and the side length of a regular tetrahedron, we know $x=\frac{3}{\sqrt{6}} \times 4=2 \sqrt{6}$.